10. Dedicated Signal Transforms

Question 1

Which assumptions are made about the course of the signal outside the frame of data samples x[k] in case a DTFT, a DFT or a DCT is employed?

Answer 1

• In case a DTFT is applied, the signal is assumed to be zero outside the frame.
• In case a DTFT is applied, the signal is assumed to be zero outside the frame.
• In case of DCT, the signal is extended to an even sequence. An even signal can for instance be obtained by extending x[k] to the left with a timereversed copy of itself and by time shifting the resulting sequence by half a sample period to the right.

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Question 2

What is the advantage of the DCT with respect to the DFT and the DTFT as far as the required type of arithmetic is concerned?

Answer 2

Disadvantage of the DTFT and the DFT is that they involve calculations with complex-valued numbers.

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Question 3

Make sure you understand the formulas defining the DCT and the IDCT, which can be found in the formulary. Interpret. Observe that k and n go from 0 till N − 1.

Answer 3

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Question 4

What is the complexity of the (I)DCT proportional to?

Answer 4

In practice efficient implementation schemes are used to calculate the (I)DCT, which have a complexity proportional to N log N.

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Question 5

What is the most important application field of the DCT? Give a practical example.

Answer 5

They are well suited for signal compression applications (JPEG images).

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Question 6

What is signal compression aiming at?

Answer 6

Signal compression aims at representing a signal with as few bits as possible without (or by almost not) compromising the signal quality.

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Question 7

Explain in words why the DCT is usually better suited to compress signals than the DFT.

Answer 7

Notice that the DCT offers a better energy compaction than the DFT. The spectral coefficients of the DCT show a larger dynamic range than those of the DFT.

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Question 8

Make sure you understand the formula defining the real cepstrum transform, which can be found in the formulary. Interpret.

What is parameter m called? Keep in mind that the real cepstrum is a non-invertible transform.

Answer 8

Discrete parameter m is called quefrency.

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Question 9

Make sure you understand the formulas defining the (inverse) complex cepstrum transform, which can be found in the formulary. Interpret.

Answer 9

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Question 10

What can the cepstrum transform be used for? Give an example of a practical application.

Answer 10

The cepstrum is used
• to detect periodicities in signals
• to detect defects in gears and rotational mechanical machinery
• to characterize (seismic) echoes
• to remove echoes from signals

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Question 11

Explain which steps need to be performed to calculate the spectral envelope of a (voice) signal.
Why is one interested in this spectral envelope in some practical applications?

Answer 11

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Question 12

Explain how echo removal through cepstral liftering works.

Which steps are performed to remove the echo?

Answer 12

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Question 13

What is deconvolution? What can it be used for?

Answer 13

This process of extracting x[k] from y[k] and undoing the filtering by h[k] is called deconvolution. It finds its application e.g. in image deblurring.

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Question 14

In telecom applications cosine waves are combined with their quadrature component (sine wave) to obtain a single-sided frequency spectrum. The quadrature component can be obtained by appropriately time delaying the signal.

Why does this no longer work in discrete time or with wideband signals?

Answer 14

In case the signal is discrete, the signal can no longer be simply delayed, as in general, non-integer delays have to be applied. On top of that, if the signal has a wideband spectrum, a
frequency dependent delay is needed for each frequency component.

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Question 15

What does the Hilbert transform do? Explain in words.

Answer 15

The Hilbert transform of a discrete-time signal x[k] is obtained by adding a fixed phase shift of −90° to the frequency spectrum at positive frequencies, and a phase shift of +90° at negative frequencies. Hence, if

formula

is the so-called Hilbert transform of x[k].

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Question 16

The formula defining the Hilbert transformer hh[k] DTFT ←→ Hh(φ) can be found in the formulary. Interpret.

Answer 16

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Question 17

What is an analytic signal?
What does the frequency spectrum of an analytic signal look like?
How do you calculate the analytic signal that corresponds to x[k]?

The formula can be found in the formulary.

Answer 17

If a signal and its Hilbert transform are added in quadrature, the analytic signal xa[k] is obtained. Note that a signal xa[k] results with a single-sided frequency spectrum. Hence, the concept of analytic signal can be considered to be a discrete-time and wideband extension of exp(j(2πf0t+θ)) = cos(2πf0t+θ) + j.sin(2πf0t+θ), as discussedon slide 31.

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Question 18

What is the envelope of x[k]? How do you calculate it?

Answer 18

Note that the analytic signal xa[k] is complex valued. It can hence be expressed as a magnitude multiplied with a phase. The magnitude is called the envelope of x[k].

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Question 19

What can the Hilbert transform be used for in practice?

Answer 19

The main applications of the Hilbert transform are in telecommunications, in theoretical signal analysis and in the computation of the envelope of a signal.

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