Digital Signal Processing/Multirate Filters

Multirate Filters
There are many instances where the rate at which a signal can be processed by a particular component or module is different from the speed at which data enters that module.

Upsampling
Upsampling creates more samples in the same amount of time, typically by inserting zero-valued samples between the preexisting samples. At first blush, this may seem like an ill-conceived approach. However, if one considers that a discrete signal is already zero between the sample points, the approach begins to make more sense.

Spectral Effects
Inserting a single zero between each of the samples will cause the spectrum to replicate and fold, creating a mirror image. If the original sample rate was $$f_s$$, the new sample rate will be $$2f_s$$, and the spectrum will fold and replicate at $$\frac{f_s}{2}$$. This image can then be filtered out using a low-pass filter.

Inserting more than one zero between sample points will cause the spectrum to replicate and fold accordion-style. If N zeros are inserted, N images will be created. Again, these images can be attenuated by applying a low-pass filter.

Implementation
Typically, the low-pass filter and the upsampler are implemented as a unit, and the upsampling takes place only conceptually. Since the zero-valued samples will not contribute to the sum of products in an FIR filter, these multiplications are simply skipped. Zero times anything is zero.

Downsampling
Downsampling, or decimation is the process of discarding certain samples so that there are fewer samples in the same amount of time. Once discarded, those samples can never be replaced and error is introduced into the system. However, a downsampled system can also be processed with a slower filter, which is typically less expensive.

Spectral Effects
Downsampling causes the spectrum to spread. If the spectrum is periodic, there could be some overlapping of spectral objects, and this causes aliasing. Aliasing of this sort is typically resolved by passing the downsampled signal through a low-pass filter to help remove the overlapped areas.

Ideal Reconstructor
An ideal reconstructor can be created by having an upsampler followed directly by a downsampler.

Similarly, if a single is separated, each branch can be delayed and downsampled, and then combined together with zero loss. This enables the signal to be processed at a much slower rate then the input data rate, without losing any data.

Fractional Decimation
Upsampling and downsamping alter the size of the data set by an integer ratio of samples. In order to achieve a fractional sample rate, upsamplers and downsamplers need to be coupled together to change the data rate to a fraction of the input data rate.