Smoothing power spectrum by convolution with boxcar function

Question

I am trying to smoothing a signal's power spectrum by convolving the spectrum with a boxcar function in frequency domain. However, the result is obviously not what I expected: original two frequency spikes become three and the frequencies are different.

Could anyone possibly point out my problem? Is the problem in the idea of convolving with a boxcar or in the MATLAB code?

Thanks...

t = 0:.001:.25;
x = sin(2*pi*50*t) + sin(2*pi*120*t);

f = 1000/251*(-floor(length(t)/2):length(t)/2);

% power spec
Fy = fftshift(fft(x,length(t)));
Pyy = Fy.*conj(Fy)/length(t);
plot(f(length(t)/2:length(t)*3/4),Pyy(length(t)/2:length(t)*3/4))
title('Power spectral density')
xlabel('Frequency (Hz)')
figure 

% rectangle function
rectLen = 5;
rw = ones(rectLen, 1);
plot(rw)

% convolution in frequency domain to smooth
Fy_conv_rw = conv(Fy, rw, 'same');
Pyy_conv_rw = Fy_conv_rw.*conj(Fy_conv_rw)/length(t) / sum(abs(rw));
plot(f(length(t)/2:length(t)*3/4),Pyy_conv_rw(length(t)/2:length(t)*3/4))
title('Power spectral density smoothed')
xlabel('Frequency (Hz)')

Answer 1

1. The spectrum is periodic so you need circular convoluioint, not linear convolution
2. You are smoothing the complex linear spectrum. Due to the phase of the complex ammplitude you will get significant interference in your smoothing window. Smoothing is typically in the energy domain and the phase is discarded. Complex smoothing is fairly difficult to do.