Aliasing: Boon or Bane
22/10/2004
Summary:
- Aliasing - if a spiral moves more than half a circle in one sample,
it seems as if it is a slower spiral (village chief's cow)
- Sampling
- Sampling should be faster than twice the highest frequency -
Nyquist theorem
- Analog filtering before ADC necessary to avoid aliasing
- Periodic signals
- Filtering of periodic signal gives periodic signal
- This can be calculated using circular convolution (circular
domain - signal wrapped around on itself)
- If the filter kernel is larger than period, the filter
kernel can be wrapped around on itself and made circular
- For periodic signal, only spirals of that period (harmonic
spirals) are important
- This harmonicity and aliasing together means a finitely
many spirals are to be used - DFT
- Derivative
- Derivative is the most basic LTI operator in continuous
domain
- Relation between step response and impulse response
- Unit shift
- Most basic discrete domain LTI operator is unit delay
- Unit delay causes linear phase change in spirals
- The linear phase shift is the frequency response of the
unit delayer
- Can be composed to get multiple sample shify
- If a signal is "stretched" by putting in alternate zeros,
the Fourier transform gets repeated twice
- The spiral and its high frequency alias look the same
at the points that matter
- The spiral and its high frequency alias cancel at
the points which don't matter
- The stretch and shift theorems and linearity together
give the fast Fourier transform (FFT) -
- decimate the signal into two signals
- recursively compute their Fourier transforms
- stretch these two signals = duplicate their Fourier transforms
- shift the "odd" signal = linear phase shift in Fourier transform
- add the two signals = add the Fourier transforms
- The FFT runs in order of n log n time
- FFT is used to implement circular convolution
- transform the signal and the filter
- multiply the transforms point-wise
- transform the result back
- Circular convolution is used to implement linear convolution
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