9. Multirate Signal Processing Flashcards
What is a multirate system?
A multirate system is a digital signal processing unit in which signals coexist that
are sampled at different sampling rates.
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How can a signal be properly downsampled by an integer factor in the digital domain?
Proper decimation, i.e. downsampling by an integer factor Q comes down to preserving the frequency content of x[k] between DC and fs/(2Q) whilst avoiding the introduction of any additional distortion such as aliasing.
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What is the most straightforward way to downsample by an integer?
A downsampling unit that reduces the sampling frequency by an integer factor of Q discards Q − 1 out of Q samples and hence, keeps each Qth sample of the input signal.
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Make sure that you can reproduce the (many) time-frequency plots that are shown on slides 5–22 of LectureMultirateDSP.pdf. You are not supposed to be able to replicate the values on the vertical axes. Recall that throughout the course material signals are commonly represented in the time and in the frequency domain as continuous Fourier transform pairs, indicated as two figures with a two-sided arrow in between with the acronym ’CFT’ on top. By representing (discrete-time) signals in the continuous time domain (as sets of Dirac impulses) (properties of) the continuous Fourier transform can be applied, which is well established from a mathematical point of view (see chapter 2 and course on Signals and Systems).
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Why is an anti-aliasing filter added to the downsampling unit?
The periodic repetitions of the baseband spectrum X(f) will typically overlap such that aliasing distortion is inserted and the spectral content of X(f) between DC and fs/(2Q) is no longer preserved.
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Where do you put the anti-aliasing filter with respect to the downsampling unit?
Which type of filter do you need?
What is the cut-off frequency?
The anti-aliasing filtering is directly applied to the discrete-time signal x[k] in the digital domain. In practice, a digital lowpass filter with cut-off frequency φc = 1/(2Q) is applied to x[k] to filter out all frequency components between fs/(2Q) and fs/2. (put in front of the downsampling unit)
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How can a signal be properly upsampled by an integer factor in the digital domain?
In other words, proper interpolation, i.e. upsampling by an integer factor P comes down to preserving the frequency content of x[k] whilst setting the frequency components above fs/2 to zero + avoiding the introduction of (aliasing) distortion.
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How is unpsampling by an integer performed?
An upsampling unit that increases the sampling frequency by an integer factor of P inserts P − 1 zeros after each input sample.
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Why is an anti-imaging filter added to the upsampling unit?
?
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Where do you put the anti-imaging filter with respect to the upsampling unit?
Which type of filter do you need?
What is the cut-off frequency?
By applying a digital lowpass filter with cut-off frequency φc = 1/(2P) to z[m] all frequency components between fs/2 and f s’/2 are filtered out. In other words, all the undesired images (copies) of the baseband spectrum X(f) are removed. (It is placed right after the upsampling unit).
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How can a signal be properly resampled by a rational factor in the digital domain?
Two approaches can be followed:
- Downsample x[k] by an integer factor Q with the scheme below on slide 7 to obtain the intermediate signal z[l] sampled at fs/Q. Next, upsample z[l] by an integer factor P using the scheme of slide 16 to get y[m].
- Upsample x[k] by an integer factor P with the scheme of slide 16 to obtain the intermediate signal z[l] sampled at fs ∙ P. Next, downsample z[l] by an integer factor Q using the scheme below on slide 7 to get y[m].
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Why is an anti-aliasing/anti-imaging filter added?
Same reason as for the up- and downsampling (?)
Where do you put the anti-aliasing/anti-imaging filter with respect to the upsampling and downsampling unit?
Which type of filter do you need?
What is the cut-off frequency?
In between the up- and downsampling unit.
A digital lowpass filter with a cut-off frequency that is a factor of max(2P, 2Q) lower than sampling frequency fs ∙ P.
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Make sure you can solve a design exercise similar to the design examples shown on slides 19–22 of LectureMultirateDSP.pdf.
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What is oversampled analog-to-digital conversion?
What are the advantages over classical analog-to-digital conversion?
Which processing steps are performed?
Make sure that you can reproduce the time-frequency plots that are shown on slides 24–29 of LectureMultirateDSP.pdf. You are not supposed to be able to replicate the values on the vertical axes.
Why can an analog filter of low order be used?
Given that highly selective analog filters are sensitive to component variations filters of (fairly) low order have to be employed.
Hence, some amount of aliasing distortion will inevitably be inserted and/or frequency components in the passband of the filter close to the cut-off frequency will get attenuated (see also section 5.1.1 of the textbook). By using the techniques discussed in the first part of the
presentation this problem can be (largely) overcome.
The idea is to sample the analog signal x(t) at a sampling rate that is an integer factor of Q higher than the required sampling rate. This is called oversampling. During the oversampling a low-order (analog) anti-aliasing filter can be employed. Next, the signal is downsampled in the digital domain by a factor of Q with the scheme presented on slide 7. This will result in a signal sampled at the required sampling rate. For the downsampling a highly selective (i.e. a high-order) digital filter can be employed.
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