Convolve a chirp signal with a dirac (kronecker delta) function (1 at x=1 and x=-1 and -1 at x=0)

Hello,

So far i tried with this code :

Fs=100; % use higher sampling rate
t=-10:1/Fs:10;
h=(abs(t)-1)==0; % use kronecker delta function for discrete-time simulation
%s=sign(t);
s = frest.Chirp('Amplitude',1e-3,...
                    'FreqRange',[10 500],...
                    'NumSamples',750);
plot(s*h);

or else with this code (not a kronecker function here) :

t=-10:.01:10;
y1= frest.Chirp('Amplitude',1e-3,...
                    'FreqRange',[10 500],...
                    'NumSamples',750);
y2=zeros(size(t)); 
y2(t>=-1 & t<=1)=1;
c=conv(y1,y2);
stem(c);

What parameter represent the period and the bandwidth of the chirp signal in frest.Chirp ?

Answer 1

[Suraj Kumar]

Hi Sam,

The 'frest.Chirp' function in MATLAB is used to create a frequency response estimation input signal, specifically a chirp signal.

In the context of the 'frest.Chirp' function in MATLAB, the parameters that relate to the period and bandwidth of the chirp signal are as follows:

• Bandwidth: This parameter is 'FreqRange' which is a two-element vector [f_start, f_end] that specifies the starting and ending frequencies of the chirp signal in Hertz (Hz).
• Period: The period of a chirp signal isn't explicitly specified by a single parameter in frest.Chirp.Instead, it is determined by the combination of the total duration of the chirp and how the frequency changes over time. The duration can be inferred from the 'NumSamples' parameter and the sampling frequency.

Now to convolve the chirp signal with a Kronecker delta function, you can refer to the following steps and attached code snippets:

1. You can define h as a vector with -1 at t = 0, and 1 at t = 1/Fs and t = -1/Fs. This represents the discrete Kronecker delta function.

Fs = 100; % Sampling frequency
t = -10:1/Fs:10; % Time vector

% Kronecker delta function
h = zeros(size(t));
h(t == 0) = -1;
h(t == 1/Fs) = 1;
h(t == -1/Fs) = 1;

2. Then you can create the 'frest.Chirp' object with specified amplitude, frequency range, and number of samples. The 'generateTimeseries' function can be used to convert this object into a time-domain signal.

% chirp signal
chirpSignal = frest.Chirp('Amplitude', 1e-3, ...
                          'FreqRange', [10 500], ...
                          'NumSamples', length(t));
% Generate the time series
s = generateTimeseries(chirpSignal);
% Convert the time series to a numeric array
sData = s.Data;

3. You can use the 'conv' function to convolve the chirp signal with the Kronecker delta function. The 'same' option ensures the output is the same length as the input signal.The result can be plotted to visualize the effect.

% Perform the convolution
c = conv(sData, h, 'same');
% Plot the result
figure;
plot(t, c);
title('Convolution of Chirp Signal with Kronecker Delta');
xlabel('Time (s)');
ylabel('Amplitude');

To learn more about the "conv" or the "generateTimeseries" functions in MATLAB, please refer to the following documentation links:

https://www.mathworks.com/help/matlab/ref/conv.html

https://www.mathworks.com/help/slcontrol/ug/generatetimeseries.html

Hope this helps!