How to do FT Time shift and Time scaling properties

Currently i am trying use matlab to do the FT properties -time shift and time scaling, can someone help me in the implementation? i would like to prove that Timeshift: F[x(t-τ) ]= e^(-jwτ) F[x(t)]

Time scaling: F[x(at)]=1/(|a|) X(w/a)

Thanks.

Answer 1

[Wayne King]

 t=0:0.001:0.1-0.001;
 Fs = 1e3;
 freq1 = 100;
 x1=cos(2*pi*freq1*t);
 Delay=2;
 yp = fft(x1);
 yp = yp(1:length(x1)/2+1);
 f = 0:Fs/length(x1):500;
 yp = yp.*exp(-1i*2*pi*f*Delay*(1/Fs));
 yp = [yp conj(fliplr(yp(2:end-1)))];
 y = ifft(yp,'symmetric');
 plot(t(1:100),x1(1:100),'b');
 hold on;
 plot(t(1:100),y(1:100),'r');

[ong]

Hi Wayne, thanks for your help. I am able to observe the shift but i couldn't get the same plot result when comparing:

 D=2;
Fs=1000;
freq1=100;
t=0:0.001:1-0.001;
    x=cos(2*pi*freq1*(t-D));
    y1 = fft(x);

and

 x1=cos(2*pi*freq1*t);
 Delay=2;
 yp = fft(x1);
 yp = yp(1:length(x1)/2+1);
 f = 0:Fs/length(x1):500;
 yp = yp.*exp(-1i*2*pi*f*Delay*(1/Fs));

any reason why? Thanks.

[AYDIN KARA]

Your code just fails when sampling frequency increases.

For f = 1e4 it gives matrix dimensions must match error. Other wise if I keep your sampling frequency at 1e6

After that point, even if I increase the delay by 2000. I could not observe any delay.

Sorry but it seems your code works for one case.

Answer 2

[Wayne King]

They agree if you get the delay right. You're not delaying the signal by 2. You're trying to delay the signal by two samples, but that has to take into account the sampling interval, so you're actually delaying the signal by 0.002 seconds.

 t = 0:0.001:1-0.001;
 freq1 = 100;
 Fs = 1000;
 x1=cos(2*pi*freq1*t);
 Delay=2;
 yp = fft(x1);
 yp = yp(1:length(x1)/2+1);
 f = 0:Fs/length(x1):500;
 yp = yp.*exp(-1i*2*pi*f*Delay*(1/Fs));
 yp = [yp conj(fliplr(yp(2:end-1)))];
 yrec = ifft(yp,'symmetric');

Compare with

D=2;
Fs=1000;
freq1=100;
t=0:0.001:1-0.001;
x=cos(2*pi*freq1*(t-(D*(1/Fs))));
y1 = fft(x);
y1T = ifft(y1,'symmetric');

max(abs(y1T-yrec))

You can see the above are identical. Thank you for accepting my answer if I have helped you.

[ong]

Hi there, Thanks for the clear explanation. However, i have one more issue, does it work the same way if i m having my fft display in the angle instead of abs?

I tried having it plot in the phase domain but there is some differ in the result.

Answer 3

[Wayne King]

 n = 0:159;
 x = cos(pi/4*n);
 y = cos(pi/4*(n-2));
 xdft = fft(x);
 ydft = fft(y);
 xdft(21)
 ydft(21)

Note that 80+i0 has become 0-80i due to the predicted phase shift of e^{-i\pi/2}

Obviously, the only way to properly "prove" that theorem is mathematically.

The scaling one you have to be careful with in discrete-time because scaling doesn't work the same with a discrete variable as it does with continuous time.

[ong]

Is there anyway that i can do it in this form:

t=0:0.001:1

x1=cos(2*pi*freq1*t)

Delay=2

yp = fft(x1);

yp = yp.*exp(-j*2*pi*t*Delay);

Answer 4

[ong]

To update on this question, Wayne King provided the explanation and the steps provided are accurate. However there is one problem, instead of ifft the abs function, it was to display in the phase domain, here, the phase for the time shift properties and the function:cos(2*pi*freq1*(t-(D*(1/Fs) doesnt match.

There are error to the phase shift, anyone can help please?

Thanks.

[Irfan Ali Dahani]

https://www.researchgate.net/profile/Irfan_Dahani

visit and you will get easy made time shifting, time scaling and amplitude scaling all in one project. surely both projects will help you.

Answer 5

[SHREEVARSHINI R]

1.Write a MATLAB program to find Fourier transform of the signal Ate-btu(t)

2.Write a MATLAB program to perform amplitude scaling, time scaling and time shift on the signal x(t) = 1+t; for t=0 to 2

Answer 6

[Sk Group]

Time Shifting Property in MATLAB Complete Prove (Code + Output)

Time shifting Prove: DFT{x(n-l)} = X(K)e^(-j(2*pi/N)kl

For complete detailed post visit: https://www.swebllc.com/time-shifting-property-in-matlab-complete-prove-code-output/

[Peter Seibold]

Proposed code NOT working! Unknown function 'sigshift1'.

Answer 7

[Sk Group]

Time Scaling in MATLAB (Code + Output)

For detailed post with complete code visit: https://www.swebllc.com/time-scaling-in-matlab-code-output/