# Convolution of a non-symmetrical window function by a cosine signal in the frequency domain

A have a time signal:

The associated DFT spectrum of this signal:

The time signal can be considered as a non-symmetrical rectangular window function multiplied by a cosine signal with a frequency $$f$$ of 10 hz with an amplitude of 17.5.

Multiplying a signals in the time domain is the same as doing a convolution in the frequency domain. A rectangular window function $$w(t)$$ is just a sinc function in the frequency domain and the $$\cos(t\omega_0) = 1/2(e^{j\omega_0} + e^{-j\omega_0})$$ is a complex exponential. The FFT:

$$DFT(w(t)\cos(t*\omega_0)) <=> 1/2W(j\omega\pm\omega_0)$$

If we only look at the positive frequencies the window function $$W$$ is just shifted by $$\omega_0$$. What I know is that the amplitude of $$W$$ at 0 Hz is 1. When shifted by $$\omega_0$$ in the frequency domain I expected the the amplitude of the 10 hz cosine signal become 17.5*1/2.

But when I look at the DFT spectrum the amplitude is not 1. Why?

The frequency shift property have to work.

• Zero pad your signal before taking the FFT so that you can see a better approximation of the DTFT: You are seeing the DFT which is samples of the DTFT. I then think it will all make more sense to you. The more you zero pad, the more samples you will interpolate in your spectrum to be able to see the DTFT. – Dan Boschen Nov 15 '19 at 21:29