Sampling Theory
A brief introduction to some sampling concepts.
The process of sampling is going to introduce a sinx/x rolloff in the passband,
which is going to have to be corrected for, usually with a post filter that has
its passband appropriately equalised. This originates from the Fourier transform
of the sampling waveform (in time domain) being of a sinx/x nature in the frequency
domain.
Consider an analogue signal with some sampling.
If the sampling waveform g(t) is of the form...
multiply this by the analogue signal f(t), to give...
ie s(t)=f(t).g(t)
Taking transforms, S(w)=F(w)*G(w) where * (in this case) means convolution.
The spectrum of g(t), ie G(w) is an infinite train of pulses (envelope has sinx/x shape).ie
Spectrum of f(t) is F(w).
If convolve the above two, then will get the total sampled data spectrum, ie S(w) which looks something like
If don't sample at a high enough rate, then will get overlap of components, ie aliasing. Two examples of this are the reverse rotating wagon wheel effect in old westerns and the banding seen around Helicopter blades.
The Nyquist theorem determines the relationship between the sampling frequency and the signal bandwidth, and states that the sampling frequency must be greater than twice the highest frequency component of the signal, for effective reconstruction to be possible. ie Fs>B*2 must be obayed, otherwise will get something like.
If you're dealing with video signals, then some multiple of the colour subcarrier frequency is probably going to be used to sample each signal (YUV, RGB etc). This is usually as low as it can be, so its usually a good idea to have pre-alias filters used before any sampling is done. Reconstruction filters are used when the signal has been processed in some way and then needs to be used in an analogue form.
For broadcast Video, Elliptic function filters are normally used, due to the tight bandwidth constraints. However, these need to be amplitude/delay equalised to prevent signal distortion.
Audio signals are generally oversampled, so a linear phase filter can be used as both the anti-alias and reconstruction filter. In some situations it’s possible to get away with a simple RC circuit. Its even cheaper to have the filtering done somewhere on silicon.
Decimation/Interpolation
This is sample rate conversion, with decimation being a lowering of sample
frequency, and interpolation being an increase in sample frequency, both of
which are going to require low pass filtering to prevent aliasing. See the
classic text by Abraham Peled and Bede Liu, "Digital Signal Processing" for
further details.
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