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You can estimate the transfer function of an unknown system based on the system's measured input and output data.

In DSP System
Toolbox™, you can estimate the transfer function
of a system using the `dsp.TransferFunctionEstimator`

System
object™ in MATLAB^{®} and
the Discrete Transfer Function Estimator block
in Simulink^{®}. The relationship between the input *x* and
output *y* is modeled by the linear, time-invariant
transfer function *T _{xy}*. The
transfer function is the ratio of the cross power spectral density
of

$${T}_{xy}(f)=\frac{{P}_{yx}(f)}{{P}_{xx}(f)}$$

The `dsp.TransferFunctionEstimator`

object and Discrete
Transfer Function Estimator block use the Welch’s averaged
periodogram method to compute the *P _{xx}* and

**Coherence**

The coherence, or magnitude-squared coherence, between *x* and *y* is
defined as:

$${C}_{xy}(f)=\frac{{\left|{P}_{xy}\right|}^{2}}{{P}_{xx}*{P}_{yy}}$$

The coherence function estimates the extent to which you can
predict *y* from *x*. The value
of the coherence is in the range 0 ≤ *C _{xy}*(

Measurements are noisy.

The system is nonlinear.

Output

*y*is a function of*x*and other inputs.

The coherence of a linear system represents the fractional part
of the output signal power that is produced by the input at that frequency.
For a particular frequency, 1 – *C _{xy}* is
an estimate of the fractional power of the output that the input does
not contribute to.

When you set the `OutputCoherence`

property
of `dsp.TransferFunctionEstimator`

to `true`

,
the object computes the output coherence. In the Discrete Transfer
Function Estimator block, to compute the coherence spectrum,
select the **Output magnitude squared coherence estimate** check
box.

To estimate the transfer function of a system in MATLAB, use the `dsp.TransferFunctionEstimator`

System object. The object implements the Welch's average modified periodogram method and uses the measured input and output data for estimation.

**Initialize the System**

The system is a cascade of two filter stages: `dsp.LowpassFilter`

and a parallel connection of `dsp.AllpassFilter`

and `dsp.AllpoleFilter`

.

allpole = dsp.AllpoleFilter; allpass = dsp.AllpassFilter; lpfilter = dsp.LowpassFilter;

**Specify Signal Source**

The input to the system is a sine wave with a frequency of 100 Hz. The sampling frequency is 44.1 kHz.

sine = dsp.SineWave('Frequency',100,'SampleRate',44100,... 'SamplesPerFrame',1024);

**Create Transfer Function Estimator**

To estimate the transfer function of the system, create the `dsp.TransferFunctionEstimator`

System object.

tfe = dsp.TransferFunctionEstimator('FrequencyRange','onesided',... 'OutputCoherence', true);

**Create Array Plot**

Initialize two `dsp.ArrayPlot`

objects: one to display the magnitude response of the system and the other to display the coherence estimate between the input and the output.

tfeplotter = dsp.ArrayPlot('PlotType','Line',... 'XLabel','Frequency (Hz)',... 'YLabel','Magnitude Response (dB)',... 'YLimits',[-120 20],... 'XOffset',0,... 'XLabel','Frequency (Hz)',... 'Title','System Transfer Function',... 'SampleIncrement',44100/1024); coherenceplotter = dsp.ArrayPlot('PlotType','Line',... 'YLimits',[0 1.2],... 'YLabel','Coherence',... 'XOffset',0,... 'XLabel','Frequency (Hz)',... 'Title','Coherence Estimate',... 'SampleIncrement',44100/1024);

By default, the *x*-axis of the array plot is in samples. To convert this axis into frequency, set the 'SampleIncrement' property of the `dsp.ArrayPlot`

object to . In this example, this value is 44100/1024, or 43.0664. For a two-sided spectrum, the `XOffset`

property of the `dsp.ArrayPlot`

object must be [-Fs/2]. The frequency varies in the range [-Fs/2 Fs/2]. In this example, the array plot shows a one-sided spectrum. Hence, set the `XOffset`

to 0. The frequency varies in the range [0 Fs/2].

**Estimate the Transfer Function**

The transfer function estimator accepts two signals: input to the two-stage filter and output of the two-stage filter. The input to the filter is a sine wave containing additive white Gaussian noise. The noise has a mean of zero and a standard deviation of 0.1. The estimator estimates the transfer function of the two-stage filter. The output of the estimator is the frequency response of the filter, which is complex. To extract the magnitude portion of this complex estimate, use the `abs`

function. To convert the result into dB, apply a conversion factor of 20*log10(magnitude).

for Iter = 1:1000 input = sine() + .1*randn(1024,1); lpfout = lpfilter(input); allpoleout = allpole(lpfout); allpassout = allpass(lpfout); output = allpoleout + allpassout; [tfeoutput,outputcoh] = tfe(input,output); tfeplotter(20*log10(abs(tfeoutput))); coherenceplotter(outputcoh); end

The first plot shows the magnitude response of the system. The second plot shows the coherence estimate between the input and output of the system. Coherence in the plot varies in the range [0 1] as expected.

**Magnitude Response of the Filter Using fvtool**

The filter is a cascade of two filter stages - `dsp.LowpassFilter`

and a parallel connection of `dsp.AllpassFilter`

and `dsp.AllpoleFilter`

. All the filter objects are used in their default state. Using the filter coefficients, derive the system transfer function and plot the frequency response using `freqz`

. Below are the coefficients in the [Num] [Den] format:

All pole filter - [1 0] [1 0.1]

All pass filter - [0.5 -1/sqrt(2) 1] [1 -1/sqrt(2) 0.5]

Lowpass filter - Determine the coefficients using the following commands:

lpf = dsp.LowpassFilter; Coefficients = coeffs(lpf);

`Coefficients.Numerator`

gives the coefficients in an array format. The mathemtical derivation of the overall system transfer function is not shown here. Once you derive the transfer function, run fvtool and you can see the frequency response below:

The magnitude response that fvtool shows matches the magnitude response that the `dsp.TransferFunctionEstimator`

object estimates.

To estimate the transfer function of a system in Simulink, use the Discrete Transfer Function Estimator block. The block implements the Welch's average modified periodogram method and uses the measured input and output data for estimation.

The system is a cascade of two filter stages: a lowpass filter and a parallel connection of an allpole filter and allpass filter. The input to the system is a sine wave containing additive white Gaussian noise. The noise has a mean of zero and a standard deviation of 0.1. The input to the estimator is the system input and the system output. The output of the estimator is the frequency response of the system, which is complex. To extract the magnitude portion of this complex estimate, use the Abs block. To convert the result into dB, the system uses a dB (1 ohm) block.

**Open and Inspect the Model**

To open the model, enter `ex_transfer_function_estimator`

in
the MATLAB command prompt.

Here are the settings of the blocks in the model.

Block | Parameter Changes | Purpose of the block |
---|---|---|

Sine Wave | **Sample time**to`1/44100` **Sample per frame**to`1024`
| Sinusoid signal with frequency at 100 Hz |

Random Source | **Source type**to`Gaussian` **Variance**to 0.01**Sample time**to 1/44100**Sample per frame**to 1024
| Random Source block generates a random noise signal with properties specified through the block dialog box |

Lowpass Filter | No change | Lowpass filter |

Allpole Filter | No change | Allpole filter with coefficients `[1 0.1]` |

Discrete Filter | **Numerator**to`[0.5 -1/sqrt(2) 1]` **Denominator**to`[1 -1/sqrt(2) 0.5]`
| Allpass filter with coefficients `[-1/sqrt(2) 0.5]` |

Discrete Transfer Function Estimator | **Frequency range**to`One-sided` **Number of spectral averages**to 8
| Transfer function estimator |

Abs | No change | Extracts the magnitude information from the output of the transfer function estimator |

First Array Plot block |
Click Select **Style**and set**Plot type**to`Line` .Select **Configuration Properties**: From the**Main**tab, set**Sample increment**to`44100/1024` and**X-offset**to`0` . In the**Display**tab, specify the**Title**as`Magnitude Response of the System in dB` ,**X-label**as`Frequency (Hz)` , and**Y-label**as`Amplitude (dB)`
| Shows the magnitude response of the system |

Second Array Plot block |
Click Select **Style**and set**Plot type**to`Line` .Select **Configuration Properties**: From the**Main**tab, set**Sample increment**to`44100/1024` and**X-offset**to`0` . In the**Display**tab, specify the**Title**as`Coherence Estimate` ,**X-label**as`Frequency (Hz)` , and**Y-label**as`Amplitude`
| Shows the coherence estimate |

By default, the *x*-axis of the array plot
is in samples. To convert this axis into frequency, the **Sample
increment** parameter is set to `Fs/1024`

.
In this example, this value is `44100/1024`

, or `43.0664`

.
For a two-sided spectrum, the **X-offset** parameter
must be `–Fs/2`

. The frequency varies in the
range `[-Fs/2 Fs/2]`

. In this example, the array
plot shows a one-sided spectrum. Hence, the **X-offset** is
set to 0. The frequency varies in the range `[0 Fs/2]`

.

**Run the Model**

The first plot shows the magnitude response of the system. The
second plot shows the coherence estimate between the input and output
of the system. Coherence in the plot varies in the range ```
[0
1]
```

as expected.

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