Many signal processing applications require changing the sampling frequency of a signal. For example, in the sub-band coding of speech and audio the digital signal sampled at Nyquist rate is split in to different frequency bands and each band is downconverted to baseband and re-sampled at a lower rate as it has lower bandwidth compared to the original signal. Each band is coded separately by quantizing with different number of bits depending on the signal amplitude. For example, Figure 12 (a) shows a signal spectrum split in to eight sub-bands and Figure 12 (b) shows the conversion of band No. 4 to baseband. All the bands two through eight are downconverted similarly. The signal in each band is re-sampled and re-quantized with different number of bits. The bandsplitting is done with digital bandpass filters. The output of filters is at the original sampling rate. Using down- sampling the sampling rate is reduced.

When there is a need to reduce the sampling rate we resort to down-sampling or decimation. When the signal is sampled at interval T we have a sequence of samples x[nT] as shown in Figure 13(a), with the corresponding spectrum shown in figure 13(b), the sampling frequency being equal to 1/T. This could be the downconverted baseband signal of Figure 12(b). As we notice the sampling rate for this signal is too large as evident from the large gap in the spectral diagram and we intend to reduce it. If we desire to reduce the sampling rate by an integer factor of M then we drop (M-1) samples and pick up every M^th sample in the original sequence. The new sequence, indexed as x[nMT], is shown in Figure 13(c). The new sampling rate is ƒ_s/ M = 1/T_M where T_M = MT. We have used M = 2 in Figure 13. The corresponding spectrum, shown in Figure 13 (d), is now being optimally utilized. It appears that we are able to decimate a sampling frequency only if the signal occupies a frequency band below ƒ_s/2M. This is partially correct. We can decimate the sampling rate even if the signal bandwidth exceeds 1/2T_M but we are interested only in frequency components below ƒ_s/2M. Before decimating the sampling rate, however, we need to use a low-pass filter, which restricts the signal band to below ƒ_s/2M Hz.

If we wish to increase the sampling rate then we do upsampling. The upsampler comprises of a sample inserter followed by a low-pass filter with a cutoff frequency L ƒ_s/2 as shown in Figure 14(a).To up-sample a signal x[nT], shown in Figure 14(b), by an integer factor L we first insert (L-1) zero-valued samples between successive original samples as in Figure 14(c) and then filter the signal. The filter interpolates the samples of proper value from the zero-valued samples as shown Figure 14(d) and, hence, is called interpolation filter.

To change the sampling rate by a rational factor K=L/M we combine interpolation and decimation techniques described above. For example, to change the sampling rate of a signal from 10 MHz to 15 MHz (K = 3/2) we upsample the signal by 3 to get 30 MHz sampling rate, then down-sample the result by two to get desired sampling rate.

Recovering the original signal from the samples is a simple and straight-forward process. All you have to do is pass the samples through a low-pass filter (bandpass filter in case of bandpass samples). How does the filter recover the signal? An ideal low-pass filter characteristics with cutoff frequency 0.5ƒ_s=1/2T (T is the sampling period) is shown in Figure 15 (a). It has a corresponding impulse response as shown in Figure 15 (b). Impulse response of a filter is the output of the filter for an impulse input. As we see when an impulse, which is very narrow pulse (ideally zero width, infinite height and finite energy) is applied to the input of the filter the output starts building up in a sinusoidal way, reaching a peak value, and then dying off in an oscillatory manner again as in Figure 15 (b). The shape of the output is mathematically defined as sinc function, sin(x)/x. The function goes through zero at instants nT, where n is a positive or negative integer. An important thing to understand is that although the position of the input impulse is shown here (and in most texts) coinciding with the zero instant of the output, in fact it is not so. How can the filter start producing an output long before input is applied? In reality, there is a delay between the input and output. For an ideal brick wall filter this delay is infinite as the tail of the output sinc function is also infinite. Also, to get a perfect brick wall frequency response for the filter we need to use infinite RC or LC sections. This is the reason why we cannot build an ideal filter, and even if we could build one, we cannot use it. The peak value of the sinc waveform is proportional to the impulse energy. Since the sampled signal is a series of impulses of height proportional to the signal value at the instant of sampling, passing them through a filter produces overlapping sinc signals as shown in Figure 15 (c). These waveforms add up to provide a resultant envelope which is the original continuous signal x(t).

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