# What are the frequency components of a sampled signal after low-pass filtering?

A continuous signal $x_a(t)$ is a linear combination of sinusoids of frequency 250 Hz, 450 Hz, 1 kHz, 2.75 kHz and 4.05 kHz. The signal $x_a(t)$ is sampled at $f_s=$1.5 kHz, and the resulting digital signal is converted into analog using a reconstruction filter with a low pass filter with cutoff frequency of 750 Hz. The resulting signals is $y_a(t)$.

The question is, which frequency components will appear in $y_a(t)$'s spectrum? I tried writing a Matlab program where I simulated $x_a(t)$ and then filtered it with a Butterworth low pass filter. I then looked at its frequency spectrum and a component of 500 Hz appeared. I showed this to my professor but he said it was wrong, and he didn't explain why.

So I'm asking you to tell me what is the real answer and why.

Thanks a lot :)

• You should not expect site members to solve your homework for you. You have a better chance of getting help if you show what you have done and ask an specific question. Now, having said that, I would suggest starting by figuring out the spectrum of the digital signal. Once you have that, the answer is straightforward. – MBaz Oct 29 '14 at 19:06
• Alright, so I added all the sinusoids in Matlab and defined a vector Y. I looked at the frequency spectrum of Y and all got was the value of 500 Hz on the plot. I believe the bandwidth of the signal is:4050-250=3.8kHz, which is a lot more than fs=1.5kHz.Therefore I think it will not be possible to recover the original signal when I convert the digital signal into an analog one (Because of Aliasing). I would like to understand why I'm getting the value of 500 Hz in the spectrum and if there is some way to predict it somehow without needing to use the Matlab code again. – Adrian Zappa Oct 29 '14 at 21:54
• There is definitely a way to find out the spectrum without Matlab. This wikipedia article on folding frequency is a good starting point. In problems like this one, Matlab is a good way to verify your answer, but it shouldn't be your main means of getting it. – MBaz Oct 29 '14 at 22:13