Overview of Multirate Filters

Multirate filters are digital filters that change the sample rate of a digital signal without introducing 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 adecimator. A filter that increases the input rate is called aninterpolator. 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.

Top graph shows a discrete time signal with a sample rate of fs. Lower graph shows the decimated version, with a sample rate of fs/2.

If you start with the top signal, sampled at a frequencyFs, then the bottom signal is sampled atFs/2 frequency. In this case, the decimation factorMis 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.

Decimator contains an anti-aliasing lowpass filter followed by a downsampler.

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

Accordiing 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 setM= 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 rangefs/2M≤f≤fs/2.
Passband must be contained in the frequency range 0 ≤f<fs/2M.
To not lose any information in the decimation process, the highest frequency of interest in the original signalfpmust be less thanfs/2M. This condition when not satisfied results in aliasing.

Effect on the Time and Frequency Domain

Consider a decimator with a downsample factor of 3.

通过输入信号x[n] through an anti-aliasing lowpass filter to yieldw[n]. The downsampler that follows reduces the filtered data by a factor of 3 by discarding 2 samples for every 3 samples to yieldy[m].

Downsampler changes the sample rate of the signal from 6 kHz to 2 kHz.

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 signalx[n] was not bandlimited by the lowpass filter. The spectrum clearly shows the role of lowpass filtering in avoiding aliasing.

左边是光谱图。右边是蒂姆e domain graph. First row of graphs correspond to the input signal x[n]. Second row of graphs show the filtered signal w[n] in both the frequency domain and the time domain. Third row of graphs show the downsampled filtered signal y[m]. Fourth row of graphs show the downsampled and unfiltered signal with 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.

Interpolator contains an upsampler followed by an anti-imaging lowpass filter.

The upsampler insertsL− 1 zero-valued samples to form the new signalw[m] at a rate ofLfs. This rate increase creates image frequencies beyond the Nyquist interval. To remove these image frequencies, pass thew[m] signal through an anti-imaging lowpass filter. The lowpass filter bandlimits thew[m] signal tofs/2 or less. The highest valid frequency after you increase the rate toLfsisLfs/2. To eliminate spectral images, use the lowpass filter to bandlimit the upsampled signal toLfs/2Lorfs/2.

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

Stopband must contain the frequency rangefs/2 ≤f≤Lfs/2.
Passband must be contained in the frequency range 0 ≤f<fs/2.
A gain ofLin the passband of the filter to compensate for the amplitude reduction by the interpolation process. Insertion ofL− 1 zeros spreads the average energy of each signal sample overLoutput samples. This effectively attenuates each sample by a factor ofL. To compensate for this attenuation, each sample of the outputy[m] needs to be multiplied byL.

Effect on the Time and Frequency Domain

Consider an interpolator with an upsample factor of 3.

通过输入信号through an upsampler. The upsampler inserts two zero-valued samples to form a new signalw[m]. Lowpass filter the signal to remove image frequencies created by the rate increase to yield they[m] signal.

Upsampler changes the sample rate from 2 kHz to 6 kHz.

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.

On left are the spectral graphs. On right are the time domain graphs. First row of graphs correspond to the input signal x[n]. Second row of graphs show the upsampled signal w[m] in both the frequency domain and the time domain. Third row of graphs show the anti-imaging filter overlapped with the upsampled signal. Fourth row of graphs show the anti-imaging filtered output.

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

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 byLand then decimating byM. 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.

Upsampler followed anti-aliasing filter, anti-imaging filter, and downsampler.

You can combine the two lowpass filters into a single filter since they are in cascade and have a common sample rate. The cascaded filterh(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 ofmin(fs/2,Lfs/2M).

Upsampler followed by a combined anti-imaging anti-aliasing lowpass filter, followed by a downsampler.

The rate converter resamples the signal according to the values ofLandM:

L<MandL/Mis not an integer –– Decimation with a noninteger conversion ratio.
L>MandL/Mis not an integer –– Interpolation with a noninteger conversion ratio.
L/M=K/1, whereKis 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.

通过输入信号through an upsampler. The upsampler inserts two zero-valued samples for each sample ofx[n]. Lowpass filter the signal to yieldv[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 ofv[i].

Upsampler changes sample rate from 2 kHz to 6 kHz. Downsampler changes the sample rate from 6 kHz to 3 kHz.

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.

左边是光谱图。右边是蒂姆e domain graph. First row of graphs correspond to the input signal x[n]. Second row of graphs show the upsampled signal w[i] in both the frequency domain and the time domain. Third row of graphs show the filtered upsampled signal v[i]. Fourth row of graphs show the signal after it is downsampled again.

当采样率转换比例很大,它is more efficient to implement the rate converter in two or more stages rather than in a single stage. For more details, seeMultistage Rate Conversion.

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卷,29号2、1981年4月,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.