Multirate signal processing is an enabling technology that brings DSP techniques to applications that require lower cost and higher sample rates. While many of the techniques have been known for more than 25 year, others have a much shorter lineage. Recent activity such as the introduction of signal-processing chip sets and design tools directed to multirate applications reflects a growing awareness of the technology. However, using multirate solutions requires a slight shift in the system designer's perspective.
The best-known consumer application is the upsampling operation that resides in the compact disk (CD) player. The audio signal on the CD is recorded as 16-bit data sampled at 44.1 kHz per channel. The playback of the audio signal through the signal-conditioning functions performed by the digital-to-analog converter (D/A) and analog low-pass filter is accomplished at four-times this sample rate, or 176.4 kHz. This upsampling is done to reduce the performance requirements, hence the cost, of the analog-smoothing filter following the D/A converter. The realization: High-speed DSP processing is less expensive than high-quality analog processing and it is sensible to trade one for the other.
Another popular consumer and commercial application is the filtering and resampling operation embedded in the sigma-delta (noise feedback) A/D converter. In a sigma-delta converter, the analog input signal is typically sampled at 64 times the signal's Nyquist rate by a 1-bit converter operating in a noise feedback loop. The feedback loop shapes the noise spectrum so that a very precise-16 bits or better-representation of the signal occupies a narrow bandwidth representing a very small fraction-say, 1/64-of the sample rate. The shaped noise spectrum occupies the complementary adjacent spectral regions.
We can easily visualize the scenario in which the converter noise is first rejected by an appropriate digital filter and then, commensurate with the bandwidth reduction related to the operation of the filter, reduces the sample rate to the signal's Nyquist rate. Similar exchanges of sample rate for amplitude resolution are employed in an upsampling sigma-delta converter. A common example is the 1-bit MASH converter used in many CD players. Here multirate filters raise the sample rate by a factor of 64 and then use a sigma-delta converter to convert the 16-bit data to 1-bit data. The D/A that follows the two transformations need only be a 1-bit converter while the smoothing filter following the D/A need only be a one-pole RC filter.
Essentially, the common factor in these examples is that in order to access certain implementation advantages, we often design and operate systems with large ratios of sample rate to bandwidth. These advantages are almost always related to cost. Occasionally, a large ratio between sample rate and bandwidth is unintentional. An example of this condition occurs when we extract a pilot tone from a composite sampled data signal such as in the composite stereo signal, or in an AMPS cellular-phone signal. A digital filter designed to extract the pilot using brute force would be prohibitively expensive to implement, while one designed to take advantage of multirate techniques has very simple, low-cost implementations. Thus, multirate signal processing must be applied whenever there is a significant ratio between the sample rate of a sampled data signal and the bandwidth of that same signal.
Basically, multirate filters are small systems, normally composed of just three building blocks. The building blocks for a down-sampling filter are a low-pass filter, a sample-rate changer such as a switch and a complex heterodyne. The low-pass filter can be recursive or non-recursive, though most designs are restricted to non-recursive for ease of access to linear phase performance.
Many systems do not make use of the complex heterodyne. We suspect that this is due to designers not being fully aware of the capabilities of multirate filters. The heterodyne function can be removed from the signal-processing chain simply by setting the index "k" to zero.
Multirate filters can be used to increase, to decrease or to preserve-with a fractional sample delay-the sample rate of a signal being processed. A filter can have two or more sample rates, for instance, an input rate and an output rate. It is possible that the filter operates internally at rates that are neither the input nor the output rate. The unique feature of the multirate-processing scheme is that the resampling operation occurs within the filter rather than after the filter for a downsampler or prior to the filter for an upsampler.
By folding the resampler into the filtering operation, the filter can be implemented in such a way that all or many of the computations occur at the lower of the two rates-input and output. When the resampling switch is embedded in the filter, the switch cycles through subsets of the prototype filter's weights. When operated in this manner, the filter exhibits a time-varying structure.
The filter is often referred to as a "periodically time varying" (PTV) circuit. Such a filter does not implement a time-invariant impulse response, but instead has a set of impulse responses related to the time of arrival of an interrogating impulse.
One effect of the periodically varying filter coefficients is that the filter itself can perform a heterodyne or frequency translation. This property is sometimes referred to as "Nyquist zone filtering." The free heterodyne is an attribute unique to multirate filters, and one for which there is no analog counterpart. A related operation might be intermediate-frequency (IF) sampling, but this is a sampling effect that does not include the filtering process.
Mathematically, multirate filters have three basic structures. These structures are the polyphase or "M-path" structure, the dyadic half-band filter formed by cascading two-path structures, and the multistage cascade integrator comb (CIC), often called the "Hogenauer form." To obtain the best combination of filtering and workload distribution the forms are often cascaded with bridging resamplers. Examples of the mixed mode may be found in the sigma-delta filters that are composed of a 16:1 fourth-order CIC followed by a 4:1 polyphase filter. For instance, Harris Semiconductor (Melbourne, Fla.) has an interesting DSP chip that uses the three structures. (See Resources for URL on application note "Programmable Down Converter, HSP 50214B.")