Smoothing in frequency domain and impulse reponse of the filter in time-domain I have a low-pass signal in frequency domain, $$\left[f_{min}=0, f_{max}=20\,\mathrm{GHz}\right]_{n=100}$$

After re-configuring the input signal as 
$$X=\left[-f_{max},\ \dots\ -f_{min},\ \mathrm{DC},\ f_{min},\ \dots\ f_{max} \right]_{2n+1}$$ 

and constructing a Hamming window of length $2n+1$ as win = hamming(2n+1),

Now the question is, how to perform smoothing and eventually get the impulse response of the filter?



Another confusing point is whether I should convolve the input signal with the window function?

-or-

Is it the multiplication between the input signal and the window function? Like X.*win?

In this context do I need perform fft(hamming(length(2n+1))) on window function?



time            = 5; % in nano-seconds
z0              = 50;
A               = importdata("sparameters_lowpass.dat");
M               = 2048;

freq            = A(:,1);
freq_neg        = -1.0*flip(freq);

realPart        = A(:,2);
imagPart        = A(:,3);

s11_freq        = realPart + imagPart*i;
s11_conj        = conj(flip(s11_freq));

F               = [freq_neg(1:end-1)', freq(1),     freq(2:end)'];
X               = [s11_conj(1:end-1)', s11_freq(1), s11_freq(2:end)']; 

win             = (hamming(length(X)));

freq_multip     = X.*win';

s11_time        = ifft(freq_multip);

z_in            = z0 * (1 + s11_time)./(1 - s11_time); % input impedance




Any help to the correct approach is very helpful.
You get the time domain values of hamming window from the hamming function. Multiplying it with the frequency domain data will not give the correct result. Smoothing is usually performed by multiplying in time domain. You could try multiplying the fourier transform of hamming window with the Frequency domain signal. and then performing the inverse fft.