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== A Representative DT Signal == | == A Representative DT Signal == | ||
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| + | For this discussion, a representative signal <math> x(t) </math> will be used to demonstrate the process of signal reconstruction. We will look at the signal in the frequency domain, as shown in the plot below: | ||
[[Image:X_f_plot.png]] | [[Image:X_f_plot.png]] | ||
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| + | As seen in the plot, the signal <math> X(f) </math> has a triangular shape and is band-limited by <math> f_{max} </math>. By the Nyquist-Shannon Sampling Theorem, the sampling frequency <math> f_s </math> must be greater than <math> 2f_{max} </math>. For this discussion, assume <math> f_s </math> is just slightly greater than <math> 2f_{max} </math>. | ||
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[[Image:Xd_w_plot.png]] | [[Image:Xd_w_plot.png]] | ||
Revision as of 17:51, 22 September 2009
Contents
Introduction
With microprocessors becoming ever increasingly faster, smaller, and cheaper, it is preferable to use digital signal processing as a way to compensate for distortions caused by analog circuitry. One area that this can be applied is in signal reconstruction, where a low pass analog filter is used on the output of a digital-to-analog converter to attenuate unwanted frequency components above the Nyquist frequency.
The problem with analog low pass filters is that higher the order, the more resistors, capacitors, and op-amps are required in its construction. More circuit components means more circuit board space, which is a precious commodity with today's hand-held devices.
Here, it will be explained how up-sampling can be used to relax requirements on analog low pass filter design while decreasing signal distortion.
A Representative DT Signal
For this discussion, a representative signal $ x(t) $ will be used to demonstrate the process of signal reconstruction. We will look at the signal in the frequency domain, as shown in the plot below:
As seen in the plot, the signal $ X(f) $ has a triangular shape and is band-limited by $ f_{max} $. By the Nyquist-Shannon Sampling Theorem, the sampling frequency $ f_s $ must be greater than $ 2f_{max} $. For this discussion, assume $ f_s $ is just slightly greater than $ 2f_{max} $.
Without Up-sampling
With Up-sampling
