Frequency response <-> Impulse response

Question

I am trying to calculate an actual frequency response from a desired frequency response.

fftSize = 128;
fs = 8000;

I have 2 input arrays: f[] and aDesired[], both of which are Real and length fftSize+1 (DC + Nyquist included). The f[] array is linearly spaced between 0 and fs/2 (0:4000). The aDesired[] array is either gains in dB or linear values.

I can easily change the length of f[] and aDesired[] as I have an interpolation function which gives any size outputs, but i thought this was a good number.

I would like to take an IFFT of ADesired[] to generate coefficients (with a configurable length), window and truncate the coeffs, and then FFT to show what the actual response would be, aActual[].

How can I take an IFFT/FFT of this data when the inputs are both Real? Or how do I prepare the data more?

Finally, it doesnt matter about the phase, if its linear or not it doesnt matter.

Is there maybe a library that could be of use with my input data?

Summary:

Input: f[129] : linearly spaced fc's
       aDesired[129] : Real values of linearly spaced gains at fc's (either in lin or log),

Processing:
       coeffs = IFFT(f, aDesired)
       coeff_new Truncate(window * coeff)
       aActual = FFT(f, coeff_new)

Output: aActual[]

Many thanks

Answer 1

As Jason R pointed out in his comment, your way of obtaining an actual frequency response is just one way of doing it. There are many other ways of determining an actual filter response. Any other frequency domain FIR filter design method could do the job, and with each method you would get a slightly different result. Having said that, let's have a look at the way you want to do it:

You have a real-valued desired response. This is the magnitude of the desired frequency response, because for causal (realizable) systems, the frequency response is always complex-valued. However, this is no problem since any finite length impulse response can always be made causal by adding some delay. First, we need to make the desired response symmetric, as is necessary for real-valued systems. Let the column vector D be the vector of desired magnitude values on the equidistant frequency grid between 0 and $f_s/2$, then the symmetric desired response is (using Matlab/Octave syntax):

Dsym=[D;D(N/2:-1:2)];

where $N$ is the FFT length. We can obtain the (non-causal) impulse response by doing an inverse FFT:

h=real(ifft(Dsym));

Note that taking the real part is only necessary to remove the tiny imaginary part due to numerical errors. To make the impulse response causal, we need to shift the second half of it to the beginning:

h_c=[h(N/2+2:N); h(1:N/2+1)];

The impulse response can now be multiplied with a window function:

h_w = h_c.*w;

Your actual frequency response is then

H=fft(h_w);

It is complex valued, so in order to compare it to the desired response, you need to plot its magnitude:

plot(abs(H))

The phase of the actual response is linear, i.e. the system just adds some delay but does not introduce any phase distortions.