# Difference in frequency spectrum of continuous time signal and discrete time signal?

I am studying chapter 4 of book signal processing first

There i came across a concept which says that(what i have understood)the spectrum of continuous time signal has limited number of lines while spectrum of discrete time signal has infinite number of lines

Why?what is the reason for this difference?

https://i.sstatic.net/3ewXv.jpg

above link Is snap of frequency spectrum diagrams of both continous time and discrete time signals

below link is relevant theory of book https://i.sstatic.net/fu8de.jpg

• You can go my answer : dsp.stackexchange.com/a/66185/49439 I have explained in detail what happens when convert Continuous signal to discrete time sequence by sampling. Commented Apr 8, 2020 at 11:38

This is the classical sampling theorem. When you sample a continuous signal $$x(t)$$, you are basically multiplying it by a sample train $$s(t) = \sum_{-\infty}^{+\infty}\delta(t-kT_s)$$, the value at consecutive $$T_s$$ being your samples. In frequency domain, it effect is to convolve $$X(\omega)$$ and $$S(\omega)$$
$$S(\omega) = \sum_{-\infty}^{+\infty}\delta(\omega-2\pi k/T_s)$$
So the resulting spectrum would be $$X_s(\omega) = \sum_{-\infty}^{+\infty}X(\omega-2\pi k/T_s)$$ There are infinite shifted copies of $$X(\omega)$$. To begin with your original signal was a single-frequency signal which traced the circumference with frequency $$f_m$$. When you sampled it at $$f_s$$ rate, you got infinite copies at $$f_m \pm kf_s$$, where $$k$$ is any integer in $$(-\infty,+\infty)$$ hence it referred to as infinite number of lines after sampling (discrete time).