How does Sampling affect a Spectrum?

In summary, sampling a signal in MATLAB with a sampling rate below the Nyquist rate and plotting the spectrum results in frequencies at +-200Hz (same as the continuous signal version) and +-4000Hz (instead of 6000Hz in the continuous signal). Sampling with an infinitesimally small window or sample and hold (also known as Zeroth order sampling) can be used to understand the effects of sampling on the spectrum of the signal, with the latter involving a sinc function in the frequency domain to account for the finite sampling time. A more in-depth explanation of this topic can be found in "Communication Systems" by Simon Haykin.
  • #1
frenzal_dude
77
0
Hi, I tried sampling [tex]
s(t)=sin(400\pi t) + 0.5cos(12000\pi t)

[/tex] at 10000Hz (below the Nyquist sampling rate)
in MATLAB and plotted the spectrum, I found that it had frequencies at +-200Hz (same as the continuous signal version) and +- 4000Hz(instead of 6000Hz in the continuous 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?

Thanks for your help,
frenzal
 
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  • #2
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.
 

1. How does the sampling rate affect the frequency range of a spectrum?

The sampling rate, or the number of samples taken per unit of time, directly affects the frequency range of a spectrum. The maximum frequency that can be accurately represented in a spectrum is half of the sampling rate. For example, if the sampling rate is 1000 samples per second, the maximum frequency that can be represented is 500 Hz.

2. How does the number of samples affect the resolution of a spectrum?

The number of samples taken also has an impact on the resolution of a spectrum. The more samples that are taken, the higher the resolution of the spectrum will be. This means that smaller changes in frequency can be detected with a higher number of samples. However, taking too many samples can also lead to a longer processing time and may not be necessary for all applications.

3. How does the sampling method affect the accuracy of a spectrum?

The sampling method used can greatly affect the accuracy of a spectrum. Different sampling methods, such as uniform or non-uniform sampling, can introduce errors or distortions in the spectrum. It is important to carefully choose the appropriate sampling method for the specific application to ensure accurate results.

4. How does the signal-to-noise ratio affect the quality of a spectrum?

The signal-to-noise ratio, or the ratio of the desired signal to the background noise, can greatly impact the quality of a spectrum. A low signal-to-noise ratio can make it difficult to distinguish between the signal and noise, leading to a lower quality spectrum. In order to improve the quality, it is important to minimize noise and increase the signal strength.

5. How does the sampling rate affect the time domain representation of a signal?

The sampling rate also affects the time domain representation of a signal. A higher sampling rate can provide a more accurate representation of the signal in the time domain, allowing for better analysis and detection of features. However, a higher sampling rate also means a larger amount of data to process and store, so a balance must be struck between the desired time domain resolution and the practical limitations of sampling rate.

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