dsp.TransferFunctionEstimator

Estimate transfer function

Description

The dsp.TransferFunctionEstimator System object™ computes the transfer function of a system, using the Welch's averaged periodogram method.

To implement the transfer function estimation object:

1. Create the dsp.TransferFunctionEstimator object and set its properties.

2. Call the object with arguments, as if it were a function.

To learn more about how System objects work, see What Are System Objects?

Creation

Syntax

tfe = dsp.TransferFunctionEstimator

tfe = dsp.TransferFunctionEstimator(Name,Value)

Description

tfe = dsp.TransferFunctionEstimator returns a transfer function estimator object, that computes the transfer function of real or complex signals. This System object uses the periodogram method and Welch’s averaged, modified periodogram method.

tfe = dsp.TransferFunctionEstimator(Name,Value) returns a transfer function estimator object with each specified property set to the specified value. Unspecified properties have default values.

Properties

Unless otherwise indicated, properties are nontunable, which means you cannot change their values after calling the object. Objects lock when you call them, and the release function unlocks them.

If a property is tunable, you can change its value at any time.

For more information on changing property values, see System Design in MATLAB Using System Objects.

FFTLengthSource — Source of FFT length value

'Auto' (default) | 'Property'

Specify the source of the FFT length value as either 'Auto' or 'Property'. If you set this property to 'Auto', the transfer function estimator sets the FFT length to the input frame size. If you set this property to 'Property', then specify the number of FFT points using the FFTLength property.

FFTLength — FFT Length

128 (default) | positive integer

Specify the length of the FFT that the transfer function estimator uses to compute spectral estimates as a positive, integer scalar.

Dependencies

This property applies when you set the FFTLengthSource property to 'Property'.

Data Types: double

Window — Window function

'Hann' (default) | 'Rectangular' | 'Chebyshev' | 'Flat Top' | 'Hamming' | 'Kaiser'

Specify a window function for the transfer function estimator as one of 'Rectangular', 'Chebyshev', 'Flat Top', 'Hamming', 'Hann', or 'Kaiser'.

SidelobeAttenuation — Side lobe attenuation of window

60 (default) | positive scalar

Specify the side lobe attenuation of the window as a real, positive scalar, in decibels (dB).

Dependencies

This property applies when you set the Window property to 'Chebyshev' or 'Kaiser'.

Data Types: double

FrequencyRange — Frequency range of the transfer function estimate

'Twosided' (default) | 'onesided' | 'centered'

Specify the frequency range of the transfer function estimator as one of 'twosided', 'onesided', or 'centered'.

If you set the FrequencyRange to 'onesided', the transfer function estimator computes the one-sided transfer function of real input signals, x and y. If the FFT length, NFFT, is even, the length of the transfer function estimate is NFFT/2+1 and is computed over the interval [0, SampleRate/2]. If NFFT is odd, the length of the transfer function estimate is equal to (NFFT+1)/2, and the interval is [0, SampleRate/2].

If FrequencyRange is set to 'twosided', the transfer function estimator computes the two-sided transfer function of complex or real input signals, x and y. The length of the transfer function estimate is equal to NFFT and is computed over [0, SampleRate].

If you set the FrequencyRange to 'centered', the transfer function estimator computes the centered two-sided transfer function of complex or real input signals, x and y. The length of the transfer function estimate is equal to NFFT and it is computed over [-SampleRate/2, SampleRate/2] for even lengths, and [-SampleRate/2, SampleRate/2] for odd lengths.

AveragingMethod — Averaging method

'Running' (default) | 'Exponential'

Specify the averaging method as 'Running' or 'Exponential'. In the running averaging method, the object computes an equally weighted average of a specified number of spectrum estimates defined by the SpectralAverages property. In the exponential method, the object computes the average over samples weighted by an exponentially decaying forgetting factor.

SpectralAverages — Number of spectral averages

8 (default) | positive integer

Specify the number of spectral averages as a positive, integer scalar. The transfer function estimator computes the current estimate by averaging the last N estimates, where N is the number of spectral averages defined in the SpectralAverages property.

Dependencies

This property applies when you set AveragingMethod to 'Running'.

Data Types: double

ForgettingFactor — Forgetting factor

0.9 (default) | scalar in the range (0,1]

Specify the exponential weighting forgetting factor as a scalar value greater than zero and smaller than or equal to one.

Tunable: Yes

Dependencies

This property applies when you set AveragingMethod to 'Exponential'.

Data Types: single | double

OutputCoherence — Magnitude squared coherence estimate

false (default) | true

Specify true to compute and output the magnitude squared coherence estimate using Welch’s averaged, modified periodogram method. The magnitude squared coherence estimate has values between 0 and 1 that indicate the correspondence at each frequency between two input signals. If you specify false, the magnitude squared coherence estimate is not computed.

Usage

Syntax

tfeEst = tfe(x,y)

[tfeEst,cxy] = tfe(x,y)

Description

tfeEst = tfe(x,y) computes the transfer function estimate, tfeEst, of the system with input x and output y using Welch's averaged periodogram method.

[tfeEst,cxy] = tfe(x,y) also computes the magnitude squared coherence estimate, cxy, of the system.

Input Arguments

x — First data input

vector | matrix

First data input, specified as a vector or a matrix. x and y must have the same size and data type.

Data Types: single | double
Complex Number Support: Yes

y — Second data input

vector | matrix

Second data input, specified as a vector or a matrix. x and y must have the same size and data type.

Data Types: single | double
Complex Number Support: Yes

Output Arguments

tfeEst — Transfer function estimate

vector | matrix

Transfer function estimate of the system for which x and y are the input and output signals, respectively.

The estimate, tfeEst, is equal to pxy./pxx, where pxy is the cross-power spectral density of x and y, and pxx is the power spectral density of x.

The transfer function estimate has the same size and data type as the input.

Data Types: single | double
Complex Number Support: Yes

cxy — Coherence estimate

vector | matrix

Magnitude squared coherence estimate of the system, returned as a vector or a matrix.

The coherence estimate, cxy, is equal to (abs(pxy).^2)./(pxx.*pyy), where pxy is the cross power spectral density of x and y, pxx is the power spectral density of x, and pyy is the power spectral density of y. For coherence to be estimated, the OutputCoherence property must be set to true.

The coherence estimate has the same size and data type as the input.

Data Types: single | double

Object Functions

To use an object function, specify the System object as the first input argument. For example, to release system resources of a System object named obj, use this syntax:

release(obj)

Specific to dsp.TransferFunctionEstimator

getFrequencyVector	Vector of frequencies at which estimation is done
getRBW	Resolution bandwidth of spectrum

Common to All System Objects

step	Run System object algorithm
release	Release resources and allow changes to System object property values and input characteristics
reset	Reset internal states of System object

Examples

Estimate Transfer Function of a System Represented by an Order-64 FIR Filter

Generate a sine wave. Use the dsp.TransferFunctionEstimator System object™ to estimate the system transfer function and the dsp.ArrayPlot System object to display it.

sin = dsp.SineWave('Frequency',100,'SampleRate',1000);
sin.SamplesPerFrame = 1000;
tfe = dsp.TransferFunctionEstimator('FrequencyRange','centered');
aplot = dsp.ArrayPlot('PlotType','Line','XOffset',-500,'YLimits',...
        [-120 5],'YLabel','Frequency Response (dB)',...
        'XLabel','Frequency (Hz)',...
        'Title','System Transfer Function');

Create an FIR Filter System object of order 64 and (normalized) cutoff frequency of 1/4. Add random noise to the sine wave. Step through the System objects to obtain the data streams, and plot the log of the magnitude of the transfer function.

firFilt = dsp.FIRFilter('Numerator',designLowpassFIR(FilterOrder=64,...
    CutoffFrequency=1/4));
for ii = 1:100
x = sin() + 0.05*randn(1000,1);
y = firFilt(x);
Txy = tfe(x,y);
aplot(20*log10(abs(Txy)))
end

Algorithms

Welch's Method of Averaged Modified Periodograms

Give two signal inputs, x and y:

1. Multiply the inputs by the window and scale the result by the window power.

2. Take FFT of the signals, X and Y.

3. Compute the current power spectral density estimates, Pxx, Pyy, and the current cross power spectral density estimate, Pyx, by taking the moving average of last N number of Z₁, Z₂, and Z₃ vectors, respectively:

   • Z₁ = X.*conj(X)

   • Z₂ = Y.*conj(Y)

   • Z₃ = Y.*conj(X)

   For details on the moving average methods, see Averaging Method.

The transfer function estimate is calculated by dividing Pyx by Pxx.

The magnitude squared coherence, Cxy, is defined by the following equation:

$$C_{xy}=\frac{\left(\text{abs}\left(P_{xy}\right).^2\right)}{\left(P{}_{xx}.*P_{yy}\right)}$$

References

[1] Hayes, Monson H. Statistical Digital Signal Processing and Modeling. Hoboken, NJ: John Wiley & Sons, 1996

[2] Kay, Steven M. Modern Spectral Estimation: Theory and Application. Englewood Cliffs, NJ: Prentice Hall, 1999

[3] Stoica, Petre and Randolph L. Moses. Spectral Analysis of Signals. Englewood Cliffs, NJ: Prentice Hall, 2005

[4] Welch, P. D. ``The use of fast Fourier transforms for the estimation of power spectra: A method based on time averaging over short modified periodograms,'' IEEE Transactions on Audio and Electroacoustics, Vol. 15, pp. 70–73, 1967.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• System Objects in MATLAB Code Generation (MATLAB Coder)

Version History

Introduced in R2013b