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F is the frequency interval between two consecutive samples of the output sequence G[ k ]. This means that T should be less than the reciprocal of 2 f H , where f H is the highest significant frequency component in the continuous time signal g t from which the sequence g[ n] was obtained. Several fast DFT algorithms require N to be an integer power of 2. A discrete-time function will have a periodic spectrum. In DFT, both the time function and frequency functions are periodic. In general, if the time-sequence is real-valued, then the DFT will have real components which are even and imaginary components that are odd.
Simi- larly, for an imaginary valued time sequence, the DFT values will have an odd real component and an even imaginary component. The FFT can be used to a obtain the power spectrum of a signal, b do digi- tal filtering, and c obtain the correlation between two signals.
The vector x is truncated or zeros are added to N, if necessary. The sampling interval is ts. Its default value is 1. The spectra are plotted versus the digital frequency F. Solution a From Equation 8. With the sampling interval being 0. The duration of g t is 0. The am- plitude of the noise and the sinusoidal signal can be changed to observe their effects on the spectrum.
Math Works Inc. Using the FFT algorithm, generate and plot the frequency content of g t. Assume a sampling rate of Hz. Find the power spectrum. Diode circuit analysis techniques will be discussed. The electronic symbol of a diode is shown in Figure 9.
Ideally, the diode conducts current in one direction. The cur- rent versus voltage characteristics of an ideal diode are shown in Figure 9. The characteristic is divided into three regions: forward-biased, reversed- biased, and the breakdown. If we assume that the voltage across the diode is greater than 0. The following example illustrates how to find n and I S from an experimental data.
Example 9. Figure 9. The thermal voltage is directly propor- tional to temperature. This is expressed in Equation 9. The reverse satura- tion current I S increases approximately 7. T1 and T2 are two different temperatures. Assuming that the emission constant of the diode is 1. We want to determine the diode current I D and the diode volt- age VD. There are several approaches for solving I D and VD.
In one approach, Equations 9. This is illustrated by the following example. Assume a temperature of 25 oC. Then, from Equation 9. Using Equation 9. The iteration technique is particularly facilitated by using computers. It consists of an alternat- ing current ac source, a diode and a resistor. The battery charging circuit, explored in the following example, consists of a source connected to a battery through a resistor and a diode.
Use MATLAB a to sketch the input voltage, b to plot the current flowing through the diode, c to calculate the conduction angle of the diode, and d calculate the peak current. Assume that the diode is ideal. The output of the half-wave rectifier circuit of Figure 9. The smoothing circuit is shown in Figure 9. When the amplitude of the source voltage VS is greater than the output volt- age, the diode conducts and the capacitor is charged.
When the source voltage becomes less than the output voltage, the diode is cut-off and the capacitor discharges with the time constant CR. The output voltage and the diode cur- rent waveforms are shown in Figure 9. Therefore, the output waveform of Figure 9. When v S t is negative, diode D1 is cut-off but diode D2 conducts.
The current flowing through the load R enters it through node A. The current entering the load resistance R enters it through node A. The output voltage of a full-wave rectifier circuit can be smoothed by connect- ing a capacitor across the load. The resulting circuit is shown in Figure 9. The output voltage and the current waveforms for the full-wave rectifier with RC filter are shown in Figure 9. The capacitor in Figure 9. Solution Peak-to-peak ripple voltage and dc output voltage can be calculated using Equations 9.
I ZM is the maximum current that can flow through the zener without being destroyed. A zener diode shunt voltage regulator circuit is shown in Fig- ure 9. Con- versely, if R is constant and VS decreases, the current flowing through the zener will decrease since the breakdown voltage is nearly constant; the output voltage will remain almost constant with changes in the source voltage VS.
Now assuming the source voltage is held constant and the load resistance is decreased, then the current I L will increase and I Z will decrease. Con- versely, ifVS is held constant and the load resistance increases, the current through the load resistance I L will decrease and the zener current I Z will increase.
In the design of zener voltage regulator circuits, it is important that the zener diode remains in the breakdown region irrespective of the changes in the load or the source voltage. From condition 1 and Equation 9. I Z ,max 9. Solution Using Thevenin Theorem, Figure 9.
In addition, when the source voltage is 35 V, the output voltage is The zener breakdown characteristics and the loadlines are shown in Figure 9. Lexton, R. Shah, M. Angelo, Jr. Sedra, A. Beards, P. Savant, Jr. Ferris, C. Ghausi, M. Warner Jr. Assume a temperature of 25 oC, emission coef- ficient, n , of 1. Both intrinsic and extrinsic semicon- ductors are discussed.
The characteristics of depletion and diffusion capaci- tance are explored through the use of example problems solved with MATLAB. The effect of doping concentration on the breakdown voltage of pn junctions is examined. Electrons surround the nucleus in specific orbits. The electrons are negatively charged and the nucleus is positively charged. If an electron absorbs energy in the form of a photon , it moves to orbits further from the nucleus.
An electron transition from a higher energy orbit to a lower energy orbit emits a photon for a direct band gap semiconductor. The energy levels of the outer electrons form energy bands. In insulators, the lower energy band valence band is completely filled and the next energy band conduction band is completely empty. The valence and conduction bands are separated by a forbidden energy gap.
In semicon- ductors the forbidden gap is less than 1. Some semiconductor materials are silicon Si , germanium Ge , and gallium arsenide GaAs. Figure Silicon has four valence electrons and its atoms are bound to- gether by covalent bonds. At absolute zero temperature the valence band is completely filled with electrons and no current flow can take place.
As the temperature of a silicon crystal is raised, there is increased probability of breaking covalent bonds and freeing electrons. The vacancies left by the freed electrons are holes. The process of creating free electron-hole pairs is called ionization. The free electrons move in the conduction band. Since electron mobility is about three times that of hole mobility in silicon, the electron current is considerably more than the hole current. The following ex- ample illustrates the dependence of electron concentration on temperature.
Solution From Equation The width of energy gap with temperature is given as . An n-type semiconductor is formed by doping the silicon crystal with elements of group V of the periodic table antimony, arse- nic, and phosphorus. The impurity atom is called a donor.
The majority car- riers are electrons and the minority carriers are holes. A p-type semiconductor is formed by doping the silicon crystal with elements of group III of the peri- odic table aluminum, boron, gallium, and indium. The impurity atoms are called acceptor atoms. The majority carriers are holes and minority carriers are electrons.
The law of mass action enables us to calculate the majority and minority car- rier density in an extrinsic semiconductor material. In an n-type semiconductor, the donor concentration is greater than the intrin- sic electron concentration, i. Example It is used to describe the energy level of the electronic state at which an electron has the probability of 0.
Equation In addition, the Fermi energy can be thought of as the average energy of mobile carriers in a semiconductor mate- rial. In an n-type semiconductor, there is a shift of the Fermi level towards the edge of the conduction band. The upward shift is dependent on how much the doped electron density has exceeded the intrinsic value.
Drift current is caused by the application of an elec- tric field, whereas diffusion current is obtained when there is a net flow of car- riers from a region of high concentration to a region of low concentration. This is shown in Figure Practical pn junctions are formed by diffusing into an n-type semiconductor a p-type impurity atom, or vice versa.
Because the p-type semiconductor has many free holes and the n-type semiconductor has many free electrons, there is a strong tendency for the holes to diffuse from the p-type to the n-type semi- conductors. Similarly, electrons diffuse from the n-type to the p-type material. When holes cross the junction into the n-type material, they recombine with the free electrons in the n-type. Similarly, when electrons cross the junction into the p-type region, they recombine with free holes.
In the junction a transition region or depletion region is created. In the depletion region, the free holes and electrons are many magnitudes lower than holes in p-type material and electrons in the n-type material. As electrons and holes recombine in the transition region, the region near the junc- tion within the n-type semiconductor is left with a net positive charge. The re- gion near the junction within the p-type material will be left with a net negative charge.
This is illustrated in Figure Because of the positive and negative fixed ions at the transition region, an elec- tric field is established across the junction. The electric field creates a poten- tial difference across the junction, the potential barrier. The potential barrier pre- vents the flow of majority carriers across the junction under equilibrium condi- tions.
That is, from Figure Typically, VC is from 0. For germanium, VC is ap- proximately 0. When a positive voltage VS is applied to the p-side of the junction and n-side is grounded, holes are pushed from the p-type material into the transition re- gion. In addition, electrons are attracted to transition region.
The depletion region decreases, and the effective contact potential is reduced. This allows majority carriers to flow through the depletion region. The depletion region increases and it become more difficult for the majority carriers to flow across the junction.
The current flow is mainly due to the flow of minority carriers. Using Equations The following example shows how I S is affected by tem- perature. During device fabrication, a p-n junction can be formed using process such as ion-implantation diffusion or epitaxy.
The dop- ing profile at the junction can take several shapes. Two popular doping pro- files are abrupt step junction and linearly graded junction. In the abrupt junction, the doping of the depletion region on either side of the metallurgical junction is a constant. This gives rise to constant charge densi- ties on either side of the junction. If the doping density on one side of the metallurgical junction is greater than that on the other side i. This condition is termed the one-sided step junction approximation.
This is the practical model for shallow junctions formed by a heavily doped diffusion into a lightly doped region of opposite polarity . In a linearly graded junction, the ionized doping charge density varies linearly across the depletion region. The charge density passes through zero at the metallurgical junction.
It can be obtained from Equations Equations The positive voltage, VC , is the contact potential of the pn junction. However, when the pn junction becomes slightly forward biased, the capacitance increases rapidly. It is also used to plot the depletion ca- pacitance. The holes are momentarily stored in the n-type material before they recombine with the majority carriers electrons in the n-type material.
Similarly, electrons are injected into and temporarily stored in the p-type material. The electrons then recombine with the majority carriers holes in the p-type material. The diffusion capacitance is usually larger than the depletion capacitance [1, 6]. Typical values of Cd ranges from 80 to pF. A small signal model of the diode is shown in Figure Cd rd Rs Cj Figure RS is the semiconductor bulk and contact resistance.
The model of the diode is shown in Figure Cj Rs Rd Figure The diffusion capacitance is zero. The resistance Rd is reverse resistance of the pn junction normally in the mega-ohms range. At a critical field, E crit , the accelerated carriers in the depletion region have sufficient energy to create new electron-hole pairs as they collide with other atoms.
The secondary electrons can in turn create more carriers in the depletion region. This is termed the avalanche breakdown process. For silicon with an impurity concentration of cm-3, the critical electric field is about 2. This high electric field is able to strip electrons away from the outer orbit of the silicon atoms, thus cre- ating hole-electron pairs in the depletion region.
This mechanism of break- down is called zener breakdown. This breakdown mechanism does not involve any multiplication effect. Normally, when the breakdown voltage is less than 6V, the mechanism is zener breakdown process. For breakdown voltages be- yond 6V, the mechanism is generally an avalanche breakdown process.
For an abrupt junction, where one side is heavily doped, the electrical proper- ties of the junction are determined by the lightly doped side. The following example shows the effect of doping concentration on breakdown voltage. Solution Using Equation Impurity Concentration' axis [1.
Singh, J. Jacoboni, C. Mousty, F. Caughey, D. IEEE, Vol. Hodges, D. Neudeck, G. II, Addison-Wesley, Beadle, W. McFarlane, G. Sze, S. Plot the difference between of ni for Equations Assume donor concentrations from to Use impurity gradient values from to It can be used to perform the basic mathematical operations: addition, subtrac- tion, multiplication, and division. They can also be used to do integration and differentiation. There are several electronic circuits that use an op amp as an integral element.
Some of these circuits are amplifiers, filters, oscillators, and flip-flops. In this chapter, the basic properties of op amps will be discussed. The non-ideal characteristics of the op amp will be illustrated, whenever possi- ble, with example problems solved using MATLAB. Its symbol is shown in Figure It is a differ- ence amplifier, with output equal to the amplified difference of the two inputs.
It also has a very large input resistance to ohms. The out- put resistance might be in the range of 50 to ohms. The offset voltage is small but finite and the frequency response will deviate considerably from the infinite frequency response. The common-mode rejection ratio is not infinite but finite. Table This condi- tion is termed the concept of the virtual short circuit. In addition, because of the large input resistance of the op amp, the latter is assumed to take no cur- rent for most calculations.
Thus, Equation R2 R1 Vin Vo Figure With the assumptions of very large open-loop gain and high input resistance, the closed-loop gain of the inverting amplifier depends on the external com- ponents R1 , R2 , and is independent of the open-loop gain. The integrating time con- stant is CR1. It behaves as a lowpass filter, passing low frequencies and at- tenuating high frequencies. However, at dc the capacitor becomes open cir- cuited and there is no longer a negative feedback from the output to the input.
The output voltage then saturates. To provide finite closed-loop gain at dc, a resistance R2 is connected in parallel with the capacitor. The circuit is shown in Figure The resistance R2 is chosen such that R2 is greater than R.
From Equation This circuit is shown in Figure For Figure The input impedance of the amplifier Z IN approaches infinity, since the current that flows into the posi- tive input of the op-amp is almost zero. R2 R1 Vo Vin Figure In addition, from Equation Plot the closed-loop gain as the open-loop gain increases from to The pole of the voltage amplifier and level shifter is KHz and has a gain of The pole of the output stage is KHz and the gain is 0.
Sketch the magnitude response of the operational amplifier open-loop gain. This causes the op amp to have a single pole lowpass response. The process of making one pole dominant in the open-loop gain characteristics is called frequency compensation, and the latter is done to ensure the stability of the op amp.
For an op amp connected in an inverting configuration Figure Slew rate is important when an output signal must follow a large input signal that is rapidly changing. If the slew rate is lower than the rate of change of the input signal, then the output voltage will be distorted. The output voltage will become triangular, and attenuated.
However, if the slew rate is higher than the rate of change of the input signal, no distortion occurs and input and output of the op amp circuit will have similar wave shapes. As mentioned in the Section In addition, the op amp has a limited output current capability, due to the saturation of the input stage. The latter is the frequency at which a sinusoidal rated output signal begins to show distortion due to slew rate limiting.
The fol- lowing example illustrates the relationship between the rated output voltage and the full-power bandwidth. CMRR decreases as frequency increases. For an inverting amplifier as shown in Figure Thus, the common-mode input voltage is approximately zero and Equation A method normally used to take into account the effect of finite CMRR in cal- culating the closed-loop gain is as follows: The contribution of the output voltage due to the common-mode input is AcmVi ,cm.
This output voltage con- tribution can be obtained if a differential input signal, Verror , is applied to the input of an op amp with zero common-mode gain. Schilling, D. Wait, J. Irvine, R. The resistance values are in kilohms.
A square wave of zero dc voltage and a peak voltage of 1 V and a frequency of KHz is connected to the input of the unity gain follower. Assume the following values of the open-loop 3 5 7 9 gain: 10 , 10 , 10 and The operation of the BJT depends on the flow of both majority and minority carriers.
There are two types of BJT: npn and pnp transistors. The electronic symbols of the two types of transistors are shown in Figure The model is shown in Figure In the case of a pnp transistor, the directions of the diodes in Figure In addition, the voltage polarities of Equations The four regions of operations are forward-active, reverse-active, saturation and cut-off.
Forward-Active Region The forward-active region corresponds to forward biasing the emitter-base junction and reverse biasing the base-collector junction. It is the normal operational region of transistors employed for amplifications.
The cut-off region corresponds to reverse biasing the base-emitter and base- collector junctions. The collector and base currents are very small compared to those that flow when transistors are in the active-forward and saturation regions. Solution From Equations Assume a temperature of oK.
The variation on V BE with temperature is similar to the changes of the pn junction diode voltage with temperature. The collector-to-base leakage current, I CBO , approximately doubles every 10o temperature rise. As discussed in Section 9.
The change in collector current can be obtained using partial derivatives. The derivation is assisted by referring to Figure Calculate the collector current at 25 oC and plot the change in collector current for temperatures between 25 and oC. At each temperature, the stability factors are calculated using Equations The change in I C for each temperature is calculated using Equation It is uneconomical to fabricate IC resistors since they take a disproportionately large area on an IC chip.
In addition, it is almost impossible to fabricate IC inductors. Biasing of ICs is done using mostly transistors that are connected to create constant current sources. Examples of integrated circuit biasing schemes are discussed in this section. The current mirror consists of two matched transistors Q1 and Q2 with their bases and emitters connected. The transistor Q1 is connected as a diode by shorting the base to its collector. In the latter mode of transistor operation, the device Q2 behaves as a current source.
To obtain an expression for the output current, we assume that all three transistors are identical. Solution We use Equation Similarly, we use Equation The input resistance is medium and is essentially independent of the load resistance RL. The output resistance is relatively high and is essentially independent of the source resistance. The bypass capacitance C E is used to increase the midband gain, since it effectively short circuits the emitter resistance R E at midband frequencies.
The resistance R E is needed for bias stability. The internal capacitances of the transistor will influence the high frequency cut-off. Solution Using Equations The zero of the overall amplifier gain is calculated using Equation The terminals of the device are the gate, source, drain, and substrate. There are two types of mosfets: the enhancement type and the depletion type. In the enhancement type MOSFET, the channel between the source and drain has to be induced by applying a voltage on the gate.
In the depletion type mosfet, the structure of the device is such that there exists a channel between the source and drain. Because of the oxide insulation between the gate and the channel, mosfets have high input resistance. The electronic symbol of a mosfet is shown in Figure Because the enhancement mode mosfet is widely used, the presentation in this section will be done using an enhancement-type mosfet.
In the latter device, the channel between the drain and source has to be induced by applying a voltage between the gate and source. The voltage needed to create the channel is called the threshold voltage, VT. For an n-channel enhancement-type mosfet , VT is positive and for a p-channel device it is negative. This implies that the drain current is zero for all values of the drain-to-source voltage.
In the latter region, the device behaves as a non-linear voltage-controlled resistance. The resistances RG1 and RG 2 will define the gate voltage. The resistance RS improves operating point stability. To obtain the drain current, it is initially assumed that the device is in saturation and Equation If Equation The method is illustrated by the following example.
Solution Substituting Equation Two solutions of I D are obtained. However, only one is sensible and possible. If the device is not in saturation, then substituting Equation The circuit is normally referred to as diode-connected enhancement transistor. A circuit for generating dc currents that are constant multiples of a reference current is shown in Figure Assuming the threshold voltages of the transistors of Figure In practice, because of the finite output resistance of transistor T2, I 0 will be a function of the output voltage v 0.
Neglect channel length modulation. Using equation However, the common- source amplifier has higher input resistance than that of the common-emitter amplifier. The circuit for the common source amplifier is shown in Figure The internal capacitances of the FET will affect the high frequency response of the amplifier.
The overall gain of the common-source amplifier can be written in a form similar to Equation The midband gain, Am , is obtained from the midband equivalent circuit of the common-source amplifier. The equivalent circuit is obtained by short-circuiting all the external capacitors and open- circuiting all the internal capacitances of the FET.
The high frequency equivalent circuit of a common-source amplifier is shown in Figure The external capacitors of the common of common- source amplifier are short-circuited at high frequencies. Determine a midband gain, b the low frequency cut-off, c high frequency cut-off, and d bandwidth of the amplifier.
Geiger, R. Savant, C. A legal weight vector is always half the length of the f and a vectors; there must be exactly one weight per band. When called with a trailing 'h' or 'Hilbert' option, firpm and firls design FIR filters with odd symmetry, that is, type III for even order or type IV for odd order linear phase filters. An ideal Hilbert transformer has this anti-symmetry property and an amplitude of 1 across the entire frequency range.
Try the following approximate Hilbert transformers and plot them using FVTool:. You can find the delayed Hilbert transform of a signal x by passing it through these filters. The analytic signal corresponding to x is the complex signal that has x as its real part and the Hilbert transform of x as its imaginary part. For this FIR method an alternative to the hilbert function , you must delay x by half the filter order to create the analytic signal:.
This method does not work directly for filters of odd order, which require a noninteger delay. In this case, the hilbert function, described in Hilbert Transform , estimates the analytic signal. Alternatively, use the resample function to delay the signal by a noninteger number of samples. Differentiation of a signal in the time domain is equivalent to multiplication of the signal's Fourier transform by an imaginary ramp function.
Approximate the ideal differentiator with a delay using firpm or firls with a 'd' or 'differentiator' option:. For a type III filter, the differentiation band should stop short of the Nyquist frequency, and the amplitude vector must reflect that change to ensure the correct slope:. The ability to omit the specification of transition bands is useful in several situations. For example, it may not be clear where a rigidly defined transition band should appear if noise and signal information appear together in the same frequency band.
Similarly, it may make sense to omit the specification of transition bands if they appear only to control the results of Gibbs phenomena that appear in the filter's response. See Selesnick, Lang, and Burrus  for discussion of this method. Instead of defining passbands, stopbands, and transition regions, the CLS method accepts a cutoff frequency for the highpass, lowpass, bandpass, or bandstop cases , or passband and stopband edges for multiband cases , for the response you specify.
In this way, the CLS method defines transition regions implicitly, rather than explicitly. The key feature of the CLS method is that it enables you to define upper and lower thresholds that contain the maximum allowable ripple in the magnitude response. Given this constraint, the technique applies the least square error minimization technique over the frequency range of the filter's response, instead of over specific bands.
The error minimization includes any areas of discontinuity in the ideal, "brick wall" response. An additional benefit is that the technique enables you to specify arbitrarily small peaks resulting from the Gibbs phenomenon. For details on the calling syntax for these functions, see their reference descriptions in the Function Reference.
As an example, consider designing a filter with order 61 impulse response and cutoff frequency of 0. Further, define the upper and lower bounds that constrain the design process as:. To approach this design problem using fircls1 , use the following commands:. Note that the y -axis shown below is in Magnitude Squared. In this case, you can specify a vector of band edges and a corresponding vector of band amplitudes.
In addition, you can specify the maximum amount of ripple for each band. Weighted CLS filter design lets you design lowpass or highpass FIR filters with relative weighting of the error minimization in each band. The fircls1 function enables you to specify the passband and stopband edges for the least squares weighting function, as well as a constant k that specifies the ratio of the stopband to passband weighting.
For example, consider specifications that call for an FIR filter with impulse response order of 55 and cutoff frequency of 0. Also assume maximum allowable passband ripple of 0. In addition, add weighting requirements:. The cfirpm filter design function provides a tool for designing FIR filters with arbitrary complex responses. It differs from the other filter design functions in how the frequency response of the filter is specified: it accepts the name of a function which returns the filter response calculated over a grid of frequencies.
This capability makes cfirpm a highly versatile and powerful technique for filter design. This design technique may be used to produce nonlinear-phase FIR filters, asymmetric frequency-response filters with complex coefficients , or more symmetric filters with custom frequency responses. The design algorithm optimizes the Chebyshev or minimax error using an extended Remez-exchange algorithm for an initial estimate.
If this exchange method fails to obtain the optimal filter, the algorithm switches to an ascent-descent algorithm that takes over to finish the convergence to the optimal solution. A linear-phase multiband filter may be designed using the predefined frequency-response function multiband , as follows:.
For the specific case of a multiband filter, we can use a shorthand filter design notation similar to the syntax for firpm :. As with firpm , a vector of band edges is passed to cfirpm. This vector defines the frequency bands over which optimization is performed; note that there are two transition bands, from —0.
The filter response for this multiband filter is complex, which is expected because of the asymmetry in the frequency domain. The impulse response, which you can select from the FVTool toolbar, is shown below.
Consider the design of a tap lowpass filter with a half-Nyquist cutoff. If we specify a negative offset value to the lowpass filter design function, the group delay offset for the design is significantly less than that obtained for a standard linear-phase design. This filter design may be computed as follows:.
The y -axis is in Magnitude Squared, which you can set by right-clicking on the axis label and selecting Magnitude Squared from the menu. Now, however, the group delay is no longer flat in the passband region. To create this plot, click the Group Delay Response button on the toolbar.
If we compare this nonlinear-phase filter to a linear-phase filter that has exactly These comparisons can assist you in deciding which filter is more appropriate for a specific application. Choose a web site to get translated content where available and see local events and offers.
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Main Content. IIR Filters Digital filters with finite-duration impulse response all-zero, or FIR filters have both advantages and disadvantages compared to infinite-duration impulse response IIR filters. FIR filters have the following primary advantages: They can have exactly linear phase. They are always stable. The design methods are generally linear.
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If there are problems with the data you select, you see messages in the Results pane. For example, the Curve Fitting app ignores Infs, NaNs, and imaginary components of complex numbers in the data, and you see messages in the Results pane in these cases.
The session file contains all the fits and variables in your session and remembers your layout. Next, select a name and location for your session file with file extension. For curves, X, Y, and Weights must be matrices with the same number of elements. Matrices of the Same Size Curve Fitting app expects inputs to be the same size. If the sizes are different but the number of elements are the same, then the tool reshapes the inputs to create a fit and displays a warning in the Results pane.
The warning indicates a possible problem with your selected data. Table Data Table data means that X and Y represent the row and column headers of a table sometimes called breakpoints and the values in the table are the values of the Z output.
If the size of Z is [n,m], the tool creates a fit but first transposes Z and warns about transforming your data. If the sizes are different but the number of elements is the same, Curve Fitting app reshapes the weights and displays a warning. Troubleshooting Data Problems If there are problems with the data you select, you see messages in the Results pane.
If you see the following warning: Duplicate x-y data points detected: using average of the z values, this means that there are two or more data points where the input values x, y are the same or very close together. The default interpolant fit type needs to calculate a unique value at that point.
You do not need do anything to fix the problem, this warning is just for your information. The Curve Fitting app automatically takes the average z value of any group of points with the same xy values. Other problems with your selected data can produce the following error: Error computing Delaunay triangulation. Please try again with different data.
Some arrangements of data make it impossible for Curve Fitting app to compute a Delaunay triangulation. Three out of the four surface interpolation methods linear, cubic, and nearest require a Delaunay triangulation of the data. An example of data that can cause this error is a case where all the data lies on a straight line in x-y. In this case, Curve Fitting app cannot fit a surface to the data. You need to provide more data in order to fit a surface.
Note Data selection is disabled if you are in debug mode. Exit debug mode to change data selections. Creating Multiple Fits After you create a single fit, it can be useful to create multiple fits to compare. When you create multiple fits you can compare different fit types and settings side-by-side in the Curve Fitting app. Each additional fit appears as a new tab in the Curve Fitting app and a new row in the Table of Fits.
This copies your selections for x, y, and z from the previous fit, and any selected validation data. No fit options are changed. Use sessions to save and reload your fits. You also can right-click a fit in the Table of Fits and select Duplicate 2 Interactive Fitting Each additional fit appears as a new tab in the Curve Fitting app. Displaying Multiple Fits Simultaneously When you have created multiple fits you can compare different fit types and settings side by side in the Curve Fitting app.
You can view plots simultaneously and you can examine the goodness-of-fit statistics to compare your fits. This section describes how to compare multiple fits. To compare plots and see multiple fits simultaneously, use the layout controls at the top right of the Curve Fitting app. Alternatively, you can click Window on the menu bar to select the number and position of tiles you want to display.
A fit figure displays the fit settings, results pane and plots for a single fit. The following example shows two fit figures displayed side by side. You can see multiple fits in the session listed in the Table of Fits. The Table of Fits displays all your fits open and closed.
Double-click a fit in the Table of Fits to open or focus if already open the fit figure. Tip If you want more space to view and compare plots, as shown next, use the View menu to hide or show the Fit Settings, Fit Results, or Table of Fits panes. The goodness-of-fit statistics help you determine how well the model fits the data.
Click the table column headers to sort by statistics, name, fit type, and so on. A value closer to zero indicates a fit that is more useful for prediction. A value closer to 1 indicates that a greater proportion of variance is accounted for by the model. A value closer to 1 indicates a better fit. A value closer to 0 indicates a fit that is more useful for prediction. When you have several fits with similar goodness-of-fit statistics, look for the smallest number of coefficients to help decide which fit is best.
You must trade off the number of coefficients against the goodness of fit indicated by the statistics to avoid overfitting. You can export individual fits to the workspace for further analysis, or you can generate MATLAB code to recreate all fits and plots in your session. By generating code you can use your interactive curve fitting session to quickly assemble code for curve and surface fits and plots into useful programs.
The file includes all fits and plots in your current session. The Curve Fitting app creates and plots a default fit to X input or predictor data and Y output or response data. The default fit is a linear polynomial fit type. Observe the fit settings display Polynomial, of Degree 1. The Curve Fitting app plots the new fit. The Curve Fitting app calculates a new fit when you change fit settings because Auto fit is selected by default.
If refitting is time consuming, e. The residuals indicate that a better fit might be possible. Therefore, continue exploring various fits to the census data set. The duplicated fit contains the same data selections and fit settings. Remove repeated data points or try centering and scaling. Normalize the data by selecting the Center and scale check box. The residuals from a good fit should look random with no apparent pattern.
A pattern, such as a tendency for consecutive residuals to have the same sign, can be an indication that a better model exists. About Scaling The warning about scaling arises because the fitting procedure uses the cdate values as the basis for a matrix with very large values. The spread of the cdate values results in a scaling problem. To address this problem, you can normalize the cdate data.
Normalization scales the predictor data to improve the accuracy of the subsequent numeric computations. A way to normalize cdate is to center it at zero mean and scale it to unit standard deviation. The equivalent code is: cdate - mean cdate. However, the functional form of the data and the resulting goodness-of-fit statistics do not change. Additionally, the data is displayed in the Curve Fitting app plots using the original scale. Determining the Best Fit To determine the best fit, you should examine both the graphical and numerical fit results.
Examine the Graphical Fit Results 1 Determine the best fit by examining the graphs of the fits and residuals. To view plots for each fit in turn, double-click the fit in the Table of Fits. Therefore, it is a poor choice and you can remove the exponential fit from the candidates for best fit. The goal of fitting the census data is to extrapolate the best fit to predict future population values.
The behavior of the sixth-degree polynomial fit beyond the data range makes it a poor choice for extrapolation and you can reject this fit. Evaluate the Numerical Fit Results When you can no longer eliminate fits by examining them graphically, you should examine the numerical fit results.
The confidence bounds on the coefficients determine their accuracy. Examine the numerical fit results: 1 For each fit, view the goodness-of-fit statistics in the Results pane. Click the column headings to sort by statistics results.
The SSE statistic is the least-squares error of the fit, with a value closer to zero indicating a better fit. The adjusted R-square statistic is generally the best indicator of the fit quality when you add additional coefficients to your model. The largest SSE for exp1 indicates it is a poor fit, which you already determined by examining the fit and residuals. The lowest SSE value is associated with poly6. However, the behavior of this fit beyond the data range makes it a poor choice for extrapolation, so you already rejected this fit by examining the plots with new axis limits.
The next best SSE value is associated with the fifth-degree polynomial fit, poly5, suggesting it might be the best fit. However, the SSE and adjusted R-square values for the remaining polynomial fits are all very close to each other. Which one should you choose? Double-click a fit in the Table of Fits to open or focus if already open the fit figure and view the Results pane. Display the fifth-degree polynomial and the poly2 fit figures side by side.
Examining results side by side can help you assess fits. Compare Fits in Curve Fitting App b To change the displayed fits, click to select a fit figure and then double-click the fit to display in the Table of Fits. Do not compare normalized coefficients directly with non-normalized coefficients. Tip Use the View menu to hide the Fit Settings or Table of Fits if you want more space to view and compare plots and results, as shown next.
You can also hide the Results pane to show only plots. The bounds cross zero on the p1, p2, and p3 coefficients for the fifth-degree polynomial. This means you cannot be sure that these coefficients differ from zero. If the higher order model terms may have coefficients of zero, they are not helping with the fit, which suggests that this model overfits the census data.
Therefore, after examining both the graphical and numerical fit results, you should select poly2 as the best fit to extrapolate the census data. Note The fitted coefficients associated with the constant, linear, and quadratic terms are nearly identical for each normalized polynomial equation. However, as the polynomial degree increases, the coefficient bounds associated with the higher degree terms cross zero, which suggests overfitting. The fittedmodel is saved as a Curve Fitting Toolbox cfit object.
Saving Your Work The toolbox provides several options for saving your work. You can then use this saved information for documentation purposes, or to extend your data exploration and analysis. You can recreate your fits and plots by calling the file at the command line with your original data as input arguments. You can also call the file with new data, and automate the process of fitting multiple data sets. This data is suitable for trying various fit settings in Curve Fitting app.
To load the example data and create, compare, and export surface fits, follow these steps: 1 To load example data to use in the Curve Fitting app, enter load franke at the MATLAB command line. The variables x, y, and z appear in your workspace. The example data is generated from Franke's bivariate test function, with added noise and scaling, to create suitable data for trying various fit settings in Curve Fitting app.
For details on the Franke function, see the following paper: Franke, R. Enter cftool, or select Curve Fitting on the Apps tab. Alternatively, you can specify the variables when you enter cftool x,y,z to open Curve Fitting app if necessary and create a default fit. The Curve Fitting app plots the data points as you select variables. When you select x, y, and z, the tool automatically creates a default surface fit. The default fit is an interpolating surface that passes through the data points.
Select the Lowess fit type from the drop-down list in the Curve Fitting app. The Curve Fitting app creates a local smoothing regression fit. Enter 10 in the Span edit box. The span defines the neighboring data points the toolbox uses to determine each smoothed value. Use the validation data to help you check that your surface is a good model, by comparing it against some other data not used for fitting.
The Specify Validation Data dialog box opens. It also adds a new row to the table of fits at the bottom. Select the Center and scale check box to normalize and correct for the large difference in scales in x and y. Normalizing the surface fit removes the warning message from the Results pane. Executing this command also exports other information such as the numbers of observations and parameters, residuals, and the fitted model.
You can treat the fitted model as a function to make predictions or evaluate the surface at values of X and Y. The X-Z view is not required, but the view makes it easier to see to remove outliers. When you move the mouse cursor to the plot, it changes to a cross-hair to show you are in outlier selection mode.
Alternatively, click and drag to define a rectangle and remove all enclosed points. A removed plot point displays as a red star in the plots. Otherwise, you can click Fit to refit the surface. To return to rotation mode, click the toolbar button Outliers mode.
Compare goodness-of-fit statistics for all fits in your session to determine which is best. You can save and reload sessions to access multiple fits. In this case, your original variables still appear in the workspace. Observe that the polynomial fit figure shows both the surface and residuals plots that you created interactively in the Curve Fitting app.
For a list of methods you can use, see sfit. Fitting a Surface To programmatically fit a surface, follow the steps in this simple example: 1 Load some data. You can transform your interactive analysis of a single data set into a reusable function for command-line analysis or for batch processing of multiple data sets.
You might need to reshape your data: see prepareCurveData or prepareSurfaceData. Smoothing is used to identify major trends in the data that can assist you in choosing an appropriate family of parametric models. If a parametric model is not evident or appropriate, smoothing can be an end in itself, providing a nonparametric fit of the data.
Note Smoothing estimates the center of the distribution of the response at each predictor. It invalidates the assumption that errors in the data are independent, and so also invalidates the methods used to compute confidence and prediction intervals. Accordingly, once a parametric model is identified through smoothing, the original data should be passed to the fit function.
You specify the model by passing a string or expression to the fit function or optional with a fittype object you create with the fittype function. Fit options specify things like weights for the data, fitting methods, and low-level options for the fitting algorithm. Exclusion rules indicate which data values will be treated as outliers and excluded from the fit.
The fit function returns a cfit for curves or sfit for surfaces object that encapsulates the computed coefficients and the fit statistics. Use the following functions to work with curve and surface fits. The Curve Fitting app allows convenient, interactive use of Curve Fitting Toolbox functions, without programming. You can, however, access Curve Fitting Toolbox functions directly, and write programs that combine curve fitting functions with MATLAB functions and functions from other toolboxes.
This allows you to create a curve fitting environment that is precisely suited to your needs. Models and fits in the Curve Fitting app are managed internally as curve fitting objects. Objects are manipulated through a variety of functions called methods. You can create curve fitting objects, and apply curve fitting methods, outside of the Curve Fitting app.
Methods are functions that operate exclusively on objects of a particular class. Data types package together objects and methods so that the methods operate exclusively on objects of their own type, and not on objects of other types.
A clearly defined encapsulation of objects and methods is the goal of object-oriented programming. Methods allow you to access and modify that information. Objects capture information from a particular fit by assigning values to coefficients, confidence intervals, fit statistics, etc. Methods allow you to post-process the fit through plotting, extrapolation, integration, etc. In other words, you can apply fittype methods to both fittype and cfit objects, but cfit methods are used exclusively with cfit objects.
Similarly for sfit objects. As an example, the fittype method islinear, which determines if a model is linear or nonlinear, would apply equally well before or after a fit; that is, to both fittype and cfit objects. On the other hand, the cfit methods coeffvalues and confint, which, respectively, return fit coefficients and their confidence intervals, would make no sense if applied to a general fittype object which describes a parametric model with undetermined coefficients.
Curve fitting objects have properties that depend on their type, and also on the particulars of the model or the fit that they encapsulate. They also allow you, through methods like plot and integrate, to perform operations that uniformly process the entirety of information encapsulated in a curve fitting object.
The methods listed in the following table are available for all fittype objects, including cfit objects. Fit Type Method Description argnames Get input argument names category Get fit category coeffnames Get coefficient names dependnames Get dependent variable name feval Evaluate model at specified predictors fittype Construct fittype object 3 Programmatic Curve and Surface Fitting Fit Type Method Description formula Get formula string indepnames Get independent variable name islinear Determine if model is linear numargs Get number of input arguments numcoeffs Get number of coefficients probnames Get problem-dependent parameter names setoptions Set model fit options type Get name of model The methods listed in the following table are available exclusively for cfit objects.
Curve Fit Method Description cfit Construct cfit object coeffvalues Get coefficient values confint Get confidence intervals for fit coefficients differentiate Differentiate fit integrate Integrate fit plot Plot fit predint Get prediction intervals probvalues Get problem-dependent parameter values A complete list of methods for a curve fitting object can be obtained with the MATLAB methods command.
These additional methods are generally low-level operations used by the Curve Fitting app, and not of general interest when writing curve fitting applications. There are no global accessor methods, comparable to getfield and setfield, available for fittype objects. Access is limited to the methods listed above. This is because many of the properties of fittype objects are derived from other properties, for which you do have access. You have read access to that property through the formula method.
You also have read access to the argument names of the object, through the argnames method. You don't, however, have direct write access to the argument names, which are derived from the formula. If you want to set the argument names, set the formula. Surface Fitting Objects and Methods Surface Fitting Objects and Methods The surface fit object sfit stores the results from a surface fitting operation, making it easy to plot and analyze fits at the command line.
See sfit. One way to quickly assemble code for surface fits and plots into useful programs is to generate a file from a session in the Curve Fitting app. In this way, you can transform your interactive analysis of a single data set into a reusable function for command-line analysis or for batch processing of multiple data sets.
You can use the generated file without modification, or edit and customize the code as needed. The error represents random variations in the data that follow a specific probability distribution usually Gaussian. The variations can come from many different sources, but are always present at some level when you are dealing with measured data.
Systematic variations can also exist, but they can lead to a fitted model that does not represent the data well. The model coefficients often have physical significance. The law of radioactive decay states that the activity of a radioactive substance decays exponentially in time. However, because the data contains some error, the deterministic component of the equation cannot be determined exactly from the data. Therefore, the coefficients and half-life calculation will have some uncertainty associated with them.
If the uncertainty is acceptable, then you are done fitting the data. If the uncertainty is not acceptable, then you might have to take steps to reduce it either by collecting more data or by reducing measurement error and collecting new data and repeating the model fit.
With other problems where there is no theory to dictate a model, you might also modify the model by adding or removing terms, or substitute an entirely different model. The Curve Fitting Toolbox parametric library models are described in the following sections. What fit types can you use for curves or surfaces? Based on your selected data, the fit category list shows either curve or surface categories.
Tip If your fit has problems, messages in the Results pane help you identify better settings. Selecting Fit Settings The Curve Fitting app provides a selection of fit types and settings that you can change to try to improve your fit.
Try the defaults first, then experiment with other settings. You can try a variety of settings within a single fit figure, and you can also create multiple fits to compare. When you create multiple fits you can compare different fit types and settings side by side in the Curve Fitting app. Selecting Model Type Programmatically You can specify a library model name as a string when you call the fit function. You can also use the fittype function to construct a fittype object for a library model, and use the fittype as an input to the fit function.
When you select this option, the tool refits with the data centered and scaled, by applying the Normalize setting to the variables. At the command line, you can use Normalize as an input argument to the fitoptions function. See the fitoptions reference page. Generally, it is a good idea to normalize inputs also known as predictor data , which can alleviate numerical problems with variables of different scales.
Then, Center and scale generally improves the fit because of the great difference in scale between the two inputs. However, if your inputs are in the same units or similar scale e. When you normalize inputs with this option, the values of the fitted coefficients change when compared to the original data.
The Curve Fitting app plots use the original scale with or without the Center and scale option. To specify fit options interactively in the Curve Fitting app, click the Fit Options button to open the Fit Options dialog box. All fit categories except interpolants and smoothing splines have configurable fit options. For polynomials you can set Robust in the Curve Fitting app, without opening the Fit Options dialog box.
The fit options for the single-term exponential are shown next. The coefficient starting values and constraints are for the census data. The method is automatically selected based on the library or custom model you use. For linear models, the method is LinearLeastSquares. For nonlinear models, the method is NonlinearLeastSquares. In most cases, this is the best choice for robust fitting. The default value is The default value is 0.
The default values depend on the model. For rational, Weibull, and custom models, default values are randomly selected within the range [0,1]. For all other nonlinear library models, the starting values depend on the data set and are calculated heuristically.
See optimized starting points below. The tool only uses the bounds with the trust region fitting algorithm. The default lower bounds for most library models are -Inf, which indicates that the coefficients are unconstrained. However, a few models have finite default lower bounds. For example, Gaussians have the width parameter constrained so that it cannot be less than 0.
See default constraints below. The default upper bounds for all library models are Inf, which indicates that the coefficients are unconstrained. For more information about these fit options, see the lsqcurvefit function in the Optimization Toolbox documentation.
If the starting points are optimized, then they are calculated heuristically based on the current data set. Random starting points are defined on the interval [0,1] and linear models do not require starting points. If a model does not have constraints, the coefficients have neither a lower bound nor an upper bound. You can override the default starting points and constraints by providing your own values using the Fit Options dialog box. You use library model names as input arguments in the fit, fitoptions, and fittype functions.
Library Model Types The following tables describe the library model types for curves and surfaces. List of Library Models for Curve and Surface Fitting Library Model Types for Surfaces Description interpolant Interpolating models, including linear, nearest neighbor, cubic spline, biharmonic, and thin-plate spline interpolation.
Model Names and Equations To specify the model you want to fit, consult the following tables for a model name to use as an input argument to the fit function. The maximum for both i and j is five. The degree of the polynomial is the maximum of i and j. The degree of x in each term will be less than or equal to i, and the degree of y in each term will be less than or equal to j. See the following table for some example model names and equations, of many potential examples. Model names are ratij, where i is the degree of the numerator and j is the degree of the denominator.
The degrees go up to five for both the numerator and the denominator. Spline Model Names Description cubicspline Cubic interpolating spline smoothingspline Smoothing spline Interpolant Model Names Type Interpolant Model Names Description Curves and Surfaces linearinterp Linear interpolation nearestinterp Nearest neighbor interpolation cubicinterp Cubic spline interpolation Curves only pchipinterp Shape-preserving piecewise cubic Hermite pchip interpolation Surfaces only biharmonicinterp Biharmonic MATLAB griddata interpolation thinplateinterp Thin-plate spline interpolation Lowess Model Names Lowess models are supported for surface fitting, not for curve fitting.
The order gives the number of coefficients to be fit, and the degree gives the highest power of the predictor variable. In this guide, polynomials are described in terms of their degree. You can use the polynomial model for interpolation or extrapolation, or to characterize data using a global fit.
For example, the temperature-to-voltage conversion for a Type J thermocouple in the 0 to o temperature range is described by a seventh-degree polynomial. Note If you do not require a global parametric fit and want to maximize the flexibility of the fit, piecewise polynomials might provide the best approach.
The main advantages of polynomial fits include reasonable flexibility for data that is not too complicated, and they are linear, which means the fitting process is simple. The main disadvantage is that high-degree fits can become unstable. Additionally, polynomials of any degree can provide a good fit within the data range, but can diverge wildly outside that range.
Therefore, exercise caution when extrapolating with polynomials. When you fit with high-degree polynomials, the fitting procedure uses the predictor values as the basis for a matrix with very large values, which can result in scaling problems. To handle this, you should normalize the data by centering it at zero mean and scaling it to unit standard deviation.
Normalize data by selecting the Center and scale check box in the Curve Fitting app. Alternatively, click Curve Fitting on the Apps tab. Change the model type from Interpolant to Polynomial. For curves, the Polynomial model fits a polynomial in x. For surfaces, the Polynomial model fits a polynomial in x and y.
The degree of the polynomial is the maximum of x and y degrees. For details, see Robust on the fitoptions reference page. You can exclude any term by setting its bounds to 0. Look in the Results pane to see the model terms, the values of the coefficients, and the goodnessof-fit statistics.
Messages in the Results pane prompt you when scaling might improve your fit. Fit Polynomials Using the Fit Function This example shows how to use the fit function to fit polynomials to data. The steps fit and plot polynomial curves and a surface, specify fit options, return goodness of fit statistics, calculate predictions, and show confidence intervals.
The polynomial library model is an input argument to the fit and fittype functions. Specify the model type poly followed by the degree in x up to 9 , or x and y up to 5. For example, you specify a quadratic curve with 'poly2' , or a cubic surface with 'poly33'.
Specify a quadratic, or second-degree polynomial, with the string 'poly2'. Fit the cubic polynomial with both center and scale and robust fitting options. Robust 'on' is a shortcut equivalent to 'Bisquare' , the default method for robust linear least-squares fitting method.
To plot a fit over a different range, set the xlimits of the axes before plotting the fit. For example, to see values extrapolated from the fit, set the upper x-limit to Polynomial models have the Method property value LinearLeastSquares, and the additional fit options properties shown in the next table. For details on all fit options, see the fitoptions reference page.
Values are 'on', 'off', 'LAR', or 'Bisquare'. The default is 'off'. Lower A vector of lower bounds on the coefficients to be fitted. The default value is an empty vector, indicating that the fit is unconstrained by lower bounds.
If bounds are specified, the vector length must equal the number of coefficients. Individual unconstrained lower bounds can be specified by Inf. Upper A vector of upper bounds on the coefficients to be fitted. The default value is an empty vector, indicating that the fit is unconstrained by upper bounds. Individual unconstrained upper bounds can be specified by Inf.
Defining Polynomial Terms for Polynomial Surface Fits You can control the terms to include in the polynomial surface model by specifying the degrees for the x and y inputs. If i is the degree in x and j is the degree in y, the total degree of the polynomial is the maximum of i and j.
The degree of x in each term is less than or equal to i, and the degree of y in each term is less than or equal to j. The model terms follow the form in this table. In this example, terms such as x3y and x2y2 are excluded because their degrees sum to more than 3. In both cases, the total degree is 4.
If the coefficient is positive, y represents exponential growth. For example, a single radioactive decay mode of a nuclide is described by a one-term exponential. If two decay modes exist, then you must use the two-term exponential model.
For the second decay mode, you add another exponential term to the model. Examples of exponential growth include contagious diseases for which a cure is unavailable, and biological populations whose growth is uninhibited by predation, environmental factors, and so on. Curve Fitting app creates the default curve fit, Polynomial. The toolbox calculates optimized start points for exponential fits, based on the current data set.
You can override the start points and specify your own values in the Fit Options dialog box. Fit Exponential Models Using the fit Function This example shows how to fit an exponential model to data using the fit function. The exponential library model is an input argument to the fit and fittype functions. Specify the model type 'exp1' or 'exp2'.
Fit a Single-Term Exponential Model Generate data with an exponential trend and then fit the data using a single-term exponential. Plot the fit and data. You can override the start points and specify your own values. Find the order of the entries for coefficients in the first model f by using the coeffnames function. Set arbitrary start points for coefficients a and b for example purposes.
For details on these options, see the table of properties for NonlinearLeastSquares on the fitoptions reference page. It is represented in either the trigonometric form or the exponential form. Fit Fourier Models Using the fit Function This example shows how to use the fit function to fit a Fourier model to data. The Fourier library model is an input argument to the fit and fittype functions.
Specify the model type fourier followed by the number of terms, e. This difference drives the trade winds in the southern hemisphere. Use Fourier series models to look for periodicity. For linear terms, you cannot be sure that these coefficients differ from zero, so they are not helping with the fit.
This means that this two term model is probably no better than a one term model. Measure Period The w term is a measure of period. Observe this looks correct on the plot, with peaks approximately 12 months apart. Examine Terms Look for the coefficients with the largest magnitude to find the most important terms. This is stronger than the 7 year cycle because the a2 and b2 coefficients have larger magnitude than a1 and b1.
Typically, the El Nino warming happens at irregular intervals of two to seven years, and lasts nine months to two years. The average period length is five years. The model results reflect some of these periods. Set Start Points The toolbox calculates optimized start points for Fourier fits, based on the current data set. Fourier series models are particularly sensitive to starting points, and the optimized values might be accurate for only a few terms in the associated equations. After examining the terms and plots, it looks like a 4 year cycle might be present.
Try to confirm this by setting w. Gaussian peaks are encountered in many areas of science and engineering. For example, Gaussian peaks can describe line emission spectra and chemical concentration assays. The toolbox calculates optimized start points for Gaussian models, based on the current data set. Fit Gaussian Models Using the fit Function This example shows how to use the fit function to fit a Gaussian model to data.
The Gaussian library model is an input argument to the fit and fittype functions. Specify the model type gauss followed by the number of terms, e. For example, the rate at which reactants are consumed in a chemical reaction is generally proportional to the concentration of the reactant raised to some power. The toolbox calculates optimized start points for power series models, based on the current data set. The power series library model is an input argument to the fit and fittype functions. Specify the model type 'power1' or 'power2'.
Note that the coefficient associated with xm is always 1. This makes the numerator and denominator unique when the polynomial degrees are the same. The main advantage of rationals is their flexibility with data that has a complicated structure.
The main disadvantage is that they become unstable when the denominator is around 0. The numerator can have degree 0 to 5, and the denominator from 1 to 5. The toolbox calculates random start points for rational models, defined on the interval [0,1]. Selecting a Rational Fit at the Command Line Specify the model type ratij, where i is the degree of the numerator polynomial and j is the degree of the denominator polynomial. For example, 'rat02', 'rat21' or 'rat55'. If you want to modify fit options such as coefficient starting values and constraint bounds appropriate for your data, or change algorithm settings, see the table of additional properties with NonlinearLeastSquares on the fitoptions reference page.
Example: Rational Fit This example fits measured data using a rational model. The data describes the coefficient of thermal expansion for copper as a function of temperature in degrees kelvin. For this data set, you will find the rational equation that produces the best fit. Note that the rational equations are not associated with physical parameters of the data. Instead, they provide a simple and flexible empirical model that you can use for interpolation and extrapolation.
The Curve Fitting app fits and plots the data. Select 2 for both Numerator degree and Denominator degree. Examine the data, fit, and residuals. Observe that the fit misses the data for the smallest and largest predictor values. Additionally, the residuals show a strong pattern throughout the entire data set, indicating that a better fit is possible.
Select 3 for both Numerator degree and Denominator degree. The fit exhibits several discontinuities around the zeros of the denominator. Note Your results depend on random start points and may vary from those shown. The message and numerical results indicate that the fit did not converge. Fit computation did not converge: Fitting stopped because the number of iterations or function evaluations exceeded the specified maximum. Although the message in the Results pane indicates that you might improve the fit if you increase the maximum number of iterations, a better choice at this stage of the fitting process is to use a different rational equation because the current fit contains several discontinuities.
These discontinuities are due to the function blowing up at predictor values that correspond to the zeros of the denominator. Select 2 for the Denominator degree and leave the Numerator degree set to 3. The data, fit, and residuals are shown below.
Therefore, you can confidently use this fit for further analysis. The main difference is that the sum of sines equation includes the phase constant, and does not include a constant intercept term. The toolbox calculates optimized start points for sum of sine models, based on the current data set. The default lower bounds for most library models are -Inf. Curve Fitting Toolbox does not fit Weibull probability distributions to a sample of data.
Instead, it fits curves to response and predictor data such that the curve has the same shape as a Weibull distribution. There are no fit settings to configure. Optional Click Fit Options to specify coefficient starting values and constraint bounds, or change algorithm settings. Fitting requires a parametric model that relates the response data to the predictor data with one or more coefficients.
The result of the fitting process is an estimate of the model coefficients. To obtain the coefficient estimates, the least-squares method minimizes the summed square of residuals. Although the least-squares fitting method does not assume normally distributed errors when calculating parameter estimates, the method works best for data that does not contain a large number of random errors with extreme values.
The normal distribution is one of the probability distributions in which extreme random errors are uncommon. However, statistical results such as confidence and prediction bounds do require normally distributed errors for their validity. If the mean of the errors is zero, then the errors are purely random. If the mean is not zero, then it might be that the model is not the right choice for your data, or the errors are not purely random and contain systematic errors.
Data that has the same variance is sometimes said to be of equal quality. The assumption that the random errors have constant variance is not implicit to weighted leastsquares regression. Instead, it is assumed that the weights provided in the fitting procedure correctly indicate the differing levels of quality present in the data.
The weights are then used to adjust the amount of influence each data point has on the estimates of the fitted coefficients to an appropriate level. Linear Least Squares Curve Fitting Toolbox software uses the linear least-squares method to fit a linear model to data. A linear model is defined as an equation that is linear in the coefficients. For example, polynomials are linear but Gaussians are not. To illustrate the linear least-squares fitting process, suppose you have n data points that can be modeled by a first-degree polynomial.
If n is greater than the number of unknowns, then the system of equations is overdetermined. Extending this example to a higher degree polynomial is straightforward although a bit tedious. All that is required is an additional normal equation for each linear term added to the model. Because inverting XTX can lead to unacceptable rounding errors, the backslash operator uses QR decomposition with pivoting, which is a very stable algorithm numerically. The projection matrix H is called the hat matrix, because it puts the hat on y.
If this assumption is violated, your fit might be unduly influenced by data of poor quality. To improve the fit, you can use weighted least-squares regression where an additional scale factor the weight is included in the fitting process. The weights determine how much each response value influences the final parameter estimates. A high-quality data point influences the fit more than a low-quality data point. Weighting your data is recommended if the weights are known, or if there is justification that they follow a particular form.
You can often determine whether the variances are not constant by fitting the data and plotting the residuals. In the plot shown below, the data contains replicate data of various quality and the fit is assumed to be correct. If you do not know the variances, it suffices to specify weights on a relative scale. Note that an overall variance term is estimated even when weights have been specified. In this instance, the weights define the relative weight to each point in the fit, but are not taken to specify the exact variance of each point.
For example, if each data point is the mean of several independent measurements, it might make sense to use those numbers of measurements as weights. Robust Least Squares It is usually assumed that the response errors follow a normal distribution, and that extreme values are rare. Still, extreme values called outliers do occur. The main disadvantage of least-squares fitting is its sensitivity to outliers.
Outliers have a large influence on the fit because squaring the residuals magnifies the effects of these extreme data points. To minimize the influence of outliers, you can fit your data using robust least-squares regression.
Therefore, extreme values have a lesser influence on the fit. Points near the line get full 4 Linear and Nonlinear Regression weight. Points farther from the line get reduced weight. Points that are farther from the line than would be expected by random chance get zero weight. For most cases, the bisquare weight method is preferred over LAR because it simultaneously seeks to find a curve that fits the bulk of the data using the usual least-squares approach, and it minimizes the effect of outliers.
Robust fitting with bisquare weights uses an iteratively reweighted least-squares algorithm, and follows this procedure: 1 Fit the model by weighted least squares. Otherwise, perform the next iteration of the fitting procedure by returning to the first step.
The plot shown below compares a regular linear fit with a robust fit using bisquare weights. Notice that the robust fit follows the bulk of the data and is not strongly influenced by the outliers. Nonlinear Least Squares Curve Fitting Toolbox software uses the nonlinear least-squares formulation to fit a nonlinear model to data.
A nonlinear model is defined as an equation that is nonlinear in the coefficients, or a combination of linear and nonlinear in the coefficients. For example, Gaussians, ratios of polynomials, and power functions are all nonlinear. Nonlinear models are more difficult to fit than linear models because the coefficients cannot be estimated using simple matrix techniques.
Instead, an iterative approach is required that follows these steps: 1 Start with an initial estimate for each coefficient. For some nonlinear models, a heuristic approach is provided that produces reasonable starting values. For other models, random values on the interval [0,1] are provided. The direction and magnitude of the adjustment depend on the fitting algorithm. It can solve difficult nonlinear problems more efficiently than the other algorithms and it represents an improvement over the popular Levenberg-Marquardt algorithm.
If the trust-region algorithm does not produce a reasonable fit, and you do not have coefficient constraints, you should try the Levenberg-Marquardt algorithm. You can use weights and robust fitting for nonlinear models, and the fitting process is modified accordingly. Because of the nature of the approximation process, no algorithm is foolproof for all nonlinear models, data sets, and starting points.
Therefore, if you do not achieve a reasonable fit using the default starting points, algorithm, and convergence criteria, you should experiment with different options. Because nonlinear models can be particularly sensitive to the starting points, this should be the first fit option you modify. Robust Fitting This example shows how to compare the effects of excluding outliers and robust fitting.
The example shows how to exclude outliers at an arbitrary distance greater than 1. The steps then compare removing outliers with specifying a robust fit which gives lower weight to outliers. Specify an informative legend. Library Models If the toolbox library does not contain a desired parametric equation, you can create your own custom equation.
Library models, however, offer the best chance for rapid convergence. For custom models, the toolbox chooses random default starting points on the interval [0,1]. You need to find suitable start points for custom models. Custom models use finite differencing. Linear and Nonlinear Fitting You can create custom general equations with the Custom Equation fit type. General models are nonlinear combinations of perhaps nonlinear terms. They are defined by equations that might be nonlinear in the parameters.
The custom equation fit uses the nonlinear least-squares fitting procedure. You can define a custom linear equation using the Custom Equation fit type, though the nonlinear fitting is less efficient and usually slower than linear least-squares fitting. You might need to search for suitable start points. Use the custom equation fit to define your own equations. An example custom equation appears when you select Custom Equation from the list, as shown here for curve data.
If you have surface data, the example custom equation uses both x and y. By default, the starting values are randomly selected on the interval [0,1] and are unconstrained. You might need to search for suitable start points and bounds. If you set fit options and then alter other fit settings, the app remembers your choices for lower and upper bounds and start points, if possible.
For custom equations Curve Fitting app always remembers user values, but for many library models if you change fit settings then the app automatically calculates new best values for start points or lower bounds. You can save your custom equations as part of your saved Curve Fitting app sessions. Your function can execute a number of times, both during fitting and during preprocessing before fitting. Be aware this may be time-consuming if you are using functions with side effects such as writing data to a file, or displaying diagnostic information to the Command Window.
Define a function in a file and use it to fit a curve. The function takes x data and some parameters for fitting. If expr is a string or anonymous function, then the toolbox uses a nonlinear fitting algorithm. You can define a custom linear equation in Custom Equation, but the nonlinear fitting is less efficient and usually slower than linear least-squares fitting. If you need linear least-squares fitting for custom equations, select Linear Fitting instead.
Linear models are linear combinations of perhaps nonlinear terms. They are defined by equations that are linear in the parameters. Tip If you need linear least-squares fitting for custom equations, select Linear Fitting. You can only see Linear Fitting in the model type list after you select some curve data, because Linear Fitting is for curves, not surfaces.
Curve Fitting app creates a default polynomial fit. An example equation appears when you select Linear Fitting from the list. Click Edit to change the example terms in the Edit Custom Linear Terms dialog box and define your own equation. Selecting Linear Fitting at the Command Line To use a linear fitting algorithm, specify a cell array of model terms as an input to the fit or fittype functions.
Do not include coefficients in the expressions for the terms. If there is a constant term, use '1' as the corresponding expression in the cell array. The equations use sums of Legendre polynomial terms. Consider an experiment in which MeV electrons are scattered from 12C nuclei. In the subsequent reaction, alpha particles are emitted and produce the residual nuclei 8Be.
By analyzing the number of alpha particles emitted as a function of angle, you can deduce certain information regarding the nuclear dynamics of 12C. The reaction kinematics are shown next. For information about generating Legendre polynomials, see the legendre function. For the alpha-emission data, you can directly associate the coefficients with the nuclear dynamics by invoking a theoretical model.
Additionally, the theoretical model introduces constraints for the infinite sum shown above. In particular, by considering the angular momentum of the reaction, a fourthdegree Legendre polynomial using only even terms should describe the data effectively. You use Linear Fitting instead of Custom Equation fit type, because the Legendre polynomials depend only on the predictor variable and constants.
The equation you will specify for the model is y1 x that is, the equation given at the beginning of this procedure. Look at each coefficient value and its confidence bounds in parentheses. The duplicated fit appears in a new tab. The Edit Custom Linear Terms dialog box opens. Note that the odd Legendre coefficients a1 and a3 are likely candidates for removal to simplify the fit, because their values are small and their confidence bounds contain zero.
I believe if you choose a good initial condition initial guess , the fitting tool works well. I wouldn't use the curve fitting toolbox for this, I'd use a curve-fitting function, e. Here is an example taken from something I did a while back:. Here some data with known parameters is generated, and some random noise added; the data is plotted.
The parameters [a, phi, tau] are estimated from the data and a curve with the estimated parameters plotted on top. Stack Overflow for Teams — Start collaborating and sharing organizational knowledge. Create a free Team Why Teams? Learn more. How to fit a curve to a damped sine wave in matlab Ask Question. Asked 9 years, 2 months ago. Modified 9 years, 2 months ago. Viewed 17k times. Edit 1 : Here's what I got using the "custom equation" option: Edit 2 : I've uploaded the data to pastebin in csv format where the first column is the amplitude and the second is the time.
SadStudent SadStudent 1 1 gold badge 4 4 silver badges 16 16 bronze badges. Add a comment. Sorted by: Reset to default. Highest score default Date modified newest first Date created oldest first. Good Luck. I know how to manually create some generic damped sine I just want cftool to find constants that will fit mine I did try the custom equation option but it didn't work well for me, but maybe I'm doing it wrong see the edit please — SadStudent.
I think with this result the fit will show a big result, You may reduce the damping ratio. What I see, the damping ratio is big, so the curve damps very early. You can multiply the damping ratio by a small number to force it to damp slower.
I can try it myself, if you can share a part of data..! It'd be wonderful if you could. I've uploaded the data to pastebin in csv format where the first column is the amplitude and the second is time.
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