## Overview of Multirate Filters

Multirate filters are digital filters that change the sample rate of a digital signal,
often incorporating FIR or IIR filters to mitigate aliasing or imaging in the rate
converted signal. These filters are categorized as decimators that reduce the sample
rate, interpolators that increase the sample rate, and rate converters that do a
combination of both. DSP System Toolbox™ offers several MATLAB^{®} System objects and Simulink^{®} blocks which implement decimators, interpolators, and rate converters.
Advanced filter technologies such as channelizers, channel synthesizers, two-channel
halfband filter banks, and multilevel filter banks use these filters as building
components.

### Decimation and Interpolation

A filter that reduces the input rate is called a *decimator*. A
filter that increases the input rate is called an *interpolator*.
The process of decimation reduces the sample rate by compressing the data, retaining
only the desired information. Interpolation, on the other hand, increases the sample
rate of the signal. Interpolation is useful, for example, when you need to feed a
signal to a system operating at a higher rate. To visualize this process, examine
the following figure, which illustrates the processes of interpolation and
decimation in the time domain.

If you start with the top signal, sampled at a frequency
*Fs*, then the bottom signal is sampled at
*Fs*/2 frequency. In this case, the decimation factor
*M* is 2. The process of decimation and interpolation allow
sample rate to be decreased or increased while minimizing undesirable effects of
errors such aliasing and imaging.

### Decimators

Conceptually, the decimator consists of an anti-aliasing lowpass filter followed by a downsampler. The role of the lowpass filter is to bandlimit the input signal before downsampling it, so that the Nyquist sampling theorem is satisfied.

The downsampler reduces the sample rate from *fs* to
*fs*/*M*. To prevent aliasing at the lower
rate of *fs*/*M*, the decimator uses a lowpass
filter prior to the downsampler. The lowpass filter bandlimits the input signal to
less than *fs*/2*M*. This bandlimiting operation
makes sure that the Nyquist criterion for sampling is satisfied, and therefore
guarantees near perfect reconstruction of the filtered signal
*w*[*n*]. If the Nyquist criterion is not
satisfied, aliasing occurs and the signal cannot be perfectly reconstructed.

According to the Nyquist theorem, for bandlimited signals, the sample rate must be
at least twice the bandwidth of the signal. For example, if you have a lowpass
filter with the highest frequency of 10 MHz, and a sample rate of 60 MHz, the
highest frequency that can be handled by the system without aliasing is 60/2 = 30
MHz, which is greater than 10 MHz. You can safely set *M* = 2 in
this case, since (60/2)/2 = 15 MHz, which is still greater than 10 MHz.

For the lowpass filter to be anti-aliasing, the filter must satisfy the following requirements:

Stopband must contain the frequency range

*fs*/2*M*≤*f*≤*fs*/2.Passband must be contained in the frequency range 0 ≤

*f*<*fs*/2*M*.To not lose any information in the decimation process, the highest frequency of interest in the original signal

*fp*must be less than*fs*/2*M*. This condition when not satisfied results in aliasing.

#### Effect on the Time and Frequency Domain

Consider a decimator with a downsample factor of 3.

Pass the input signal *x*[*n*] through an
anti-aliasing lowpass filter to yield
*w*[*n*]. The downsampler that follows reduces
the filtered data by a factor of 3 by discarding 2 samples for every 3 samples
to yield *y*[*m*].

In the frequency domain, the input signal with a sample rate of 6 kHz first passes through the lowpass filter. The lowpass filter bandlimits the signal and removes image frequencies, which would otherwise cause aliasing. The downsampler that follows downsamples the filtered signal by a factor of 3 to a sample rate of 2 kHz.

The spectral components in the dashed lines are the frequencies that would
have been aliased if the input signal *x*[*n*]
was not bandlimited by the lowpass filter. The spectrum clearly shows the role
of lowpass filtering in avoiding aliasing.

### Interpolators

Conceptually, an interpolator consists of an upsampler followed by an anti-imaging lowpass filter. The role of the lowpass filter is to remove the image frequencies caused by the upsampler.

The upsampler inserts *L* − 1 zero-valued samples to form the new
signal *w*[*m*] at a rate of
*Lfs*. This rate increase creates image frequencies beyond the
Nyquist interval. To remove these image frequencies, pass the
*w*[*m*] signal through an anti-imaging
lowpass filter. The lowpass filter bandlimits the
*w*[*m*] signal to *fs*/2 or
less. The highest valid frequency after you increase the rate to
*Lfs* is *Lfs*/2. To eliminate spectral
images, use the lowpass filter to bandlimit the upsampled signal to
*Lfs*/2*L* or *fs*/2.

Based on these constraints, the lowpass filter must satisfy the following filter requirements:

Stopband must contain the frequency range

*fs*/2 ≤*f*≤*Lfs*/2.Passband must be contained in the frequency range 0 ≤

*f*<*fs*/2.A gain of

*L*in the passband of the filter to compensate for the amplitude reduction by the interpolation process. Insertion of*L*− 1 zeros spreads the average energy of each signal sample over*L*output samples. This effectively attenuates each sample by a factor of*L*. To compensate for this attenuation, each sample of the output*y*[*m*] needs to be multiplied by*L*.

#### Effect on the Time and Frequency Domain

Consider an interpolator with an upsample factor of 3.

Pass the input signal through an upsampler. The upsampler inserts two
zero-valued samples to form a new signal
*w*[*m*]. Lowpass filter the signal to
remove image frequencies created by the rate increase to yield the
*y*[*m*] signal.

In the frequency domain, pass the input signal with a sample rate of 2 kHz through an upsampler. The sample rate changes to 6 kHz. The lowpass filter that follows bandlimits the signal and removes image frequencies which would otherwise cause aliasing.

The spectral components in the dashed lines are the frequencies removed by the anti-imaging lowpass filter.

For more information about the effects of decimation and interpolation on a sampled signal, see References.

### Sample Rate Converters

Sample rate converters change the sample rate of a signal by a noninteger factor,
*L*/*M*. The sample rate change is achieved by
first interpolating the data by *L* and then decimating by
*M*. The interpolation precedes decimation because in that
order the overall process is not as lossy.

Conceptually, a sample rate converter can be represented using the following diagram.

You can combine the two lowpass filters into a single filter since they are in
cascade and have a common sample rate. The cascaded filter
*h*(*k*) must be both anti-aliasing and
anti-imaging in order to effectively rate convert the signal without introducing
significant aliasing and imaging errors. The combined filter has a cutoff frequency
of `min`

(*fs*/2,
*Lfs*/2*M*).

The rate converter resamples the signal according to the values of
*L* and *M*:

*L*<*M*and*L*/*M*is not an integer –– Decimation with a noninteger conversion ratio.*L*>*M*and*L*/*M*is not an integer –– Interpolation with a noninteger conversion ratio.*L*/*M*=*K*/`1`

, where*K*is an integer –– Interpolation with an integer conversion ratio.*L*/*M*=`1`

/*K*–– Decimation with an integer conversion ratio.

#### Effect on the Time and Frequency Domain

Consider a sample rate converter with a noninteger conversion factor 3/2.

Pass the input signal through an upsampler. The upsampler inserts two
zero-valued samples for each sample of
*x*[*n*]. Lowpass filter the signal to yield
*v*[*i*]. Then pass the filtered data
through a downsampler. The downsampler reduces the signal by a factor of 2 by
retaining only one sample for every two samples of
*v*[*i*].

In the frequency domain, the input signal with a sample rate of 2 kHz is first increased by a factor of 3 to 6 kHz. The lowpass filter that follows bandlimits the signal and removes image frequencies which would otherwise cause aliasing. The filtered signal is then downsampled by a factor of 2 to 3 kHz.

The spectral components in the dashed lines are the frequencies which are removed by the lowpass filter.

When the sample rate conversion ratio is large, it is more efficient to implement the rate converter in two or more stages rather than in a single stage. For more details, see Multistage Rate Conversion.

#### Reducible Multirate Filters

A multirate filter is said to be reducible if you can modify it through noble
identities to produce a single-rate FIR or IIR filter. The equivalent filter has
a single FIR or IIR filter stage. Reducible multirate filters are useful because
you can characterize and analyze them by their single-stage FIR or IIR
equivalent. Therefore, you can call standard analysis functions such as
`freqz`

, `impz`

, and `grpdelay`

on reducible multirate
filters.

Consider this filter that is a cascade of an FIR rate converter and an FIR decimator.

redFilt = cascade(dsp.FIRRateConverter(11,3),dsp.FIRDecimator(7))

Here is a conceptual diagram. The filter contains two stages.

redFilt = dsp.FilterCascade with properties: Stage1: [1×1 dsp.FIRRateConverter] Stage2: [1×1 dsp.FIRDecimator] CloneStages: true

Applying noble identity for decimation.

Replacing the two filters *h*_{1}(k) and
*h*_{2}(3*k*) with an
equivalent filter *h*(*k*). The equivalent
filter object is `dsp.FIRRateConverter(11,21,num)`

with
```
num = conv(h1.Numerator,
upsample(h2.Numerator,3))
```

.

Alternatively, you can call `tf(redFilt)`

to obtain the
transfer function of the equivalent reduced filter.

#### Irreducible Multirate Filters

Irreducible multirate filters do not have an equivalent FIR or IIR rate conversion filter, that is, they are not reducible.

As an example, consider a cascade of an FIR decimator and an FIR interpolator.

irredFilt = cascade(dsp.FIRDecimator(2),dsp.FIRInterpolator(2))

Here is a diagrammatic representation of the filter. This filter contains two
stages. However, you cannot apply noble identities on this filter to reduce it.
The filter is therefore irreducible. To analyze irreducible filters in the
frequency domain, use the `freqzmr`

function.

## References

[1] Fliege, Norbert. *Multirate Digital Signal Processing: Multirate Systems, Filter
Banks, Wavelets*. Wiley, 1994.

[2] Harris, Fred. *Multirate Signal Processing for Communication Systems*.
Prentice Hall PTR, 2004.

[3] Hogenauer, E. “An
Economical Class of Digital Filters for Decimation and Interpolation.” *IEEE Transactions on Acoustics, Speech, and Signal
Processing*, vol. 29, no. 2, Apr. 1981, pp. 155–62.

[4] Lyons, Richard G.
*Understanding Digital Signal Processing*. 2nd
ed, Prentice Hall PIR, 2004.

[5] Mitra, S.K.
*Digital Signal Processing*, McGraw-Hill, 1998.

[6] Orfanidis, Sophocles J.
*Introduction to Signal Processing*. Prentice
Hall, 1996.