# Modulation and DeModulation

I have a question regarding Modulation and Demodulation.
In question 5.21 (b) from the "Discrete-time Signal Processing" book by Oppennheim, he asks to find out the type of filter used in the attached figure:

The question is what is the effect of $(-1)^n$ on the original filter and y[n].
I saw the answer to be high-pass filter, but I am not sure why.
Please help

Thanks

• Hint: $(-1)^n = \cos(\pi n)$. Multiplying $x(n)$ by $\cos(\pi n)$ is equivalent to which operation in the frequency domain? From this, you can sketch the effect of the modulation, the filter, and the second modulation on x's spectrum. – pichenettes Nov 18 '14 at 1:11

## 1 Answer

As hinted out, you firstly shift signal by π by multiplying it by cos(πn). Now, your low frequency components are around π. In the next step of diagram, you filter the components around frequency 0 since the filter in diagram is lowpass. But since your signal was shifted by π, you filter the high frequency components of the original signal. In the next step, you shift your original signal back to 0 frequency.

So, overall you filter the high frequency components.

• The hint and answer are great!!!! I completely understand now. The book by Oppenheim and Schafer Book give lots of interesting questions.... I have another question related. Can I assume that ( x[n] (-1)^n * hlp[n] ) = (x[n] * (-1)^n hlp[n] ) meaning a shifted signal convulsion with low pass filter is the same as the signal convulsion with high pass filter? – Mona Nov 19 '14 at 16:45