Processing for compression - The MPEG - page 69

Sampling theory states that the sampling rate needed must be at least twice the bandwidth in the signal to be
sampled. If the signal is band limited, the sampling rate need only be more than twice the signal bandwidth not the
signal frequency . Downsampled signals of this kind can be reconstructed by a reconstruction or synthesis filter
having a bandpass response rather than a low pass response. As only signals within the passband can be output,
it is clear from Figure 3.24 that the waveform which will result is the original as the intermediate aliased waveform
lies outside the passband.
Figure 3.24: The sample stream shown would ordinarily represent the waveform shown in (a), but if it is known that
the original signal could exist only between two frequencies then the waveform in (b) must be the correct one. A
suitable bandpass reconstruction filter, or synthesis filter, will produce the waveform in (b).
Figure 3.25 shows the operation of a simple QMF. At (a) the input spectrum of the PCM audio is shown, having an
audio baseband extending up to half the sampling rate and the usual lower sideband extending down from there up
to the sampling frequency. The input is passed through a FIR low-pass filter which cuts off at one quarter of the
sampling rate to give the spectrum shown at (b). The input also passes in parallel through a second FIR filter which
is physically identical, but the coefficients are different. The impulse response of the FIR LPF is multiplied by a
cosinusoidal waveform which amplitude modulates it. The resultant impulse gives the filter a frequency response
shown at (c). This is a mirror image of the LPF response. If certain criteria are met, the overall frequency response
of the two filters is flat. The spectra of both (b) and (c) show that both are oversampled by a factor of 2 because
they are half-empty. As a result both can be decimated by a factor of two, which is the equivalent of dropping every
other sample. In the case of the lower half of the spectrum, nothing remarkable happens. In the case of the upper
half, it has been resampled at half the original frequency as shown at (d). The result is that the upper half of the
audio spectrum aliases or heterodynes to the lower half.
Figure 3.25: The quadrature mirror filter. In (a) the input spectrum has an audio baseband extending up to half the
sampling rate. The input is passed through an FIR low-pass filter which cuts off at one-quarter of the sampling rate
to give the spectrum shown in (b). The input also passes in parallel through a second FIR filter whose impulse
response has been multiplied by a cosinusoldal waveform in order to amplitude-modulate it. The resultant impulse
gives the filter a mirror image frequency response shown in (c). The spectra of both (b) and (c) show that both are