Sampled and aliasing signal

Question

() = cos(200) + 2cos(320). (), is produced by sampling () at sampling frequency $f = 300 Hz$. Is the signal sampled or aliased and why?

Answer 1

Another perspective: The sampling process can be viewed as a type of amplitude modulation. PCM stands for Pulse Code Modulation. Let's set aside Code as simply the encoding of an amplitude as a digital value, leaving Pulse Modulation. This is the process of multiplying, or amplitude modulation of, the analog signal with a pulse train.

The results of amplitude modulation is well known—it produces the sum and difference frequencies of the component sinusoids of each. Amplitude modulation of 100 Hz by 9 Hz produces frequencies of 109 Hz and 91 Hz. For signals with more sinusoidal components, the result is the sum of all combinations of AM of the individual components.

So, if we know the harmonic content of a pulse train, we know the results of modulating another signal with it. The harmonic series of a pulse train in time is a pulse train in frequency. That is, a pulse train at frequency Fs (our sampling frequency) has harmonics that are cosines of equal amplitude at integer multiples of Fs.

And fortunately, the problem is made simple since we are primarily concerned with the harmonics of the pulse train that frame the band we are sampling (below half the sample rate). The first non-negative harmonics are 0 Fs and 1 Fs. For your sample rate of 300 Hz, amplitude modulation of your signal by 0 Hz and 300 Hz.

Modulation by 0 Hz (again, a cosine, of non-zero amplitude, so it's simply a constant offset and often referred to as DC) simply gives the spectrum of your signal being sampled.

Your signal has harmonics of 100 Hz and 160 Hz, so that will also be the spectral components we get from AM by DC.

Amplitude modulation by 300 Hz yields a spectrum that is the sum and difference. We can ignore the sum, since we want to know if there is aliasing in our band of interest. For modulation by 300 Hz, the difference frequencies resulting from AM with your signal will be (300 - 100) Hz, or 200 Hz, and (300 - 160) Hz, or 140 Hz. Here is the spectrum with that contrubution, in red:

Accounting for these, we have a spectrum with 100 Hz, 140 Hz, 160 Hz, and 200 Hz. When we satisfy reconstruction by lowpass filtering to pass everything below half the sample rate, or 150 Hz, we end up reproducing 100 Hz and 140 Hz, instead of the original 100 Hz and 160 Hz. Here is the spectrum of the result, reconstructed for the analog domain:

Despite the long explanation, once you know this, the visualization is easy—simply mirror the spectrum of the original signal off the sampling frequency.

Answer 2

The second cosine has a frequency of 160Hz, given the sampling freqeuency of 300Hz, there would be Aliasing. This is because sampling frequency should be greater than twice the maximum signal frequency

Edit: after additional details by OP

The sinuoid at frequency 160Hz, would be aliased to 140Hz. The sinuosid at 100Hz will have another coomponent at 200Hz, so you would have four sinusoids 100, 140Hz, 200, 160 and scaled in amplitude by T.

This happens because sampling in time domain creates periodic repetitions of the spectrum in frqeuency domain. let  denote the spectrum of the signal  then after sampling the spectrum of the sampled signal is given by the expression

Now if you look at the spectrum of a sinuosid it has two impulses one  other at , for example the sinuosid at frequency 160Hz has two impulses one at -160 and other at +160Hz, now when you plug in this spectrum in the expression above and keep getting shifted copies you will find that you get impulses at +140,+440,+740 ......and so on similarly at the negative frquencies as well.

Also the sinusoid at 100Hz will similarly get shifted copies at 100(itself, meaning  in the above expression) then at 200Hz (-100 +300) similary at -100Hz(100-300Hz).

Now the filter cutoff is at 250Hz so all these components will be seen.

The ideal sampling and filters for the signal should have been sampling rate greater than 320Hz and filter cutoff of 160Hz