How to find IDTFT of an array in matlab?

We have freqz function to find the DTFT of an array which gives the 512 samples (0 to pi) in frequency domain.

For example freqz used on rectangular window of length 45, which gave 512 complex points array in frequency domain. Now I want to reverse this operation to get the rectangular window of 45 points in time domain.

Does matlab has any function for IDTFT or atleast there is a way we can acheive this with or without MATLAB.

Also how to get (0 to 2pi) points instead if (0 to pi) points using freqz?

Answer 1

[Paul]

As @Matt J noted, you can specify the frequency vector that freqz should use to cover whatever frequencies are desired. Or freqz accepts a parameter 'whole' that makes it use a frequency vector all the way around the unit circle (except the last point is not 2*pi, it's 2*pi - dw)

As for inverting the DTFT, I don't believe that Matlab has a built-in function. But the iDTFT is defined in terms of an integral which can be evaluted numerically (or symbolically I suppose). Assuming you already have h and w, the iDTFT integrand can potentially be computed by interpolation of h.

Here is a link that may be helpful. Based on the Question, I think you want the second version in Eq. 4.

Inverse DTFT

If you try to code it up and have some problems, feel free to post a follow-up comment showing what you've done and what the problem is.

[Palguna Gopireddy]

I used below code to find th IDTFT

[r_frq,w]=freqz(rectwin(45));
r_frq2(512:1023)=r_frq;
r_frq3=flip(r_frq,1);
r_frq2(1:511)=r_frq3(1:511);

n=0:44;
syms f(w,n)
f(w)=r_frq.*exp(j*w*n);
r_time2=(1/2*pi)*int(f(w,n),w,-pi,pi);

It is saying error: Array sizes must match.

I know I am multplying r_frq with 512 elements and n is with 45 elements. But how to get the output with 45 elements with 'int' function on an array with 512 elements.

How should I proceed now. What should I do.

[Paul]

I think the problem in the code is that rfrq has 511 elements ( maybe it should have 512?) and exp(j*w*n) has only 45 elements. So those can't be elementwise multiplied. But even if the dimensions are corrected, the code still has other problems/

However, if you want to use sympolic math for finite duration sequences. there is no need to use freqz() at all, because we know the equation for the DTFT

Let's attack the problem for a simple sequence that maybe can be adapted as needed.

Instead of rectwin(45), let the sequence r(n) be defined as:

r(n) = n for 0 <= n <= 10, and 0 otherwise

rvals = 1:10; % ensure r is a row vector

Because rvals has only 10 non-nonzero elements corresponding to 1 <= n <= 10, its DTFT is simply

syms w real
R(w) = sum(rvals.*exp(-1j*(1:10)*w));

Let's compare R(w) to freqz() for some random frequencies just to make sure

rng(100)
wvals = rand(1,5)*2*pi;
abs(double(R(wvals)-freqz([0 rvals],1,wvals))) % pad with leading zero because rvals(1) corresponds to n = 1

ans = 1×5

1.0e-13 * 0.1537 0.0877 0.0513 0.2158 0.0916

Not exactly zero, but close enough in floating point.

Now that we have R(w) we can use the iDTFT integral to recover r(n)

syms n integer
r(n) = int(R(w)*exp(1j*w*n),w,0,2*sym(pi),'hold',true)/2/sym(pi);

Evaluate r(n) at some values of n

[-5:15; release(r(-5:15))]

ans = 

which looks like the expected result, i.e., n for 1 <= n <= 9 and 0 otherwise.

If use of freqz() is required, then symbolic math is probably not the way to go.

[Palguna Gopireddy]

Thanks

Answer 2

[Matt J]

Does matlab has any function for IDTFT

It doesn't because your frequency domain data is not really continuous, you're really looking for the IFFT, which MATLAB does have a function for (namely ifft() ).

Also how to get (0 to 2pi) points instead if (0 to pi) points using freqz?

From the doc:

h = freqz(___,w) returns the frequency response vector h evaluated at the normalized frequencies supplied in w.

[Paul]

Assumptions using ifft() would be

a) the points in h correspond to equally spaced points in w over the range 0 - 2pi(-), e.g., if using the 'whole' option with freqz

b) the number of points in h (and w) is equal to or greater than the length of the discrete-time sequence.

[Palguna Gopireddy]

I tried using

w=0:1:2*pi;

[h,w]=freqz(_,w);

it is giving error, saying 'Invalid text character. Check for unsupported symbol, invisible character, or pasting of non-ASCII characters'.

Basically however I use '_' inside freqz parantheis, it is giving same error.

I tried copy paste below from the document with n=45

[h,w] = freqz(___,n,'whole');

It is giving the same error.

I also tried

[h,w]= freqz(rectwin(45),'whole');

It is giving error.

Error using freqz

Expected input number 1, A, to be one of these types:

single, double

Error in freqz (line 112)

validateattributes(a,{'single','double'},{},'freqz','A',1);

So I am thinking there is no way to get 0 to 2*pi directly from the freqz.

BUt I found that using 'wvtool' we can specify range 0 to 2*pi. By I don't think there is no way to export these values into workspace. Is there?

[Star Strider]

The first argument to freqz must be one or more filter parameters, such as the numerator and denominator of a transfer function, or a second-order-section matrix.

[Palguna Gopireddy]

So In the documaentation [h,w] = freqz(___,n,'whole'); means we should not give '_' directly, we should put array of numerator and denomenator parameters in that place?