# How does Sampling affect a Spectrum?

1. Jun 4, 2010

### frenzal_dude

Hi, I tried sampling $$s(t)=sin(400\pi t) + 0.5cos(12000\pi t)$$ at 10000Hz (below the Nyquist sampling rate)
in matlab and plotted the spectrum, I found that it had frequencies at +-200Hz (same as the continous signal version) and +- 4000Hz(instead of 6000Hz in the continous signal).

Just wondering how sampling actually affects the spectrum of the signal, is there a formula or something which can tell you the frequency components based on the sampling rate?

frenzal

2. Jun 5, 2010

### mohammed.omar

Hi frenzal_dude,

Ideal Sampling (Sampling with an infinitesimally small window) is like convolving the signal in frequency domain by a series of Dirac Deltas. So you would just repeat the spectrum with a spacing = sampling frequency throughout the spectrum.

Sample and hold, also known as Zeroth order sampling, on the other hand is like the ideal sampling, but multuplied by a sinc function in the frequency domain, to account for the finite sampling time.

Let me know if you are having any problem with this. You can also take a look at "Communication Systems" by Simon Haykin; it has a very nice introduction to this topic.