Chapter 5, TRANSFORMS AND SUBBAND DECOMPOSITION Video Solutions, Multimedia Signals and Systems

01:34

Problem 1

Show that the $4 \times 4$ forward DFT transformation matrix given in Eq. (5.11) is unitary.

Victor Salazar

Numerade Educator

01:34

Problem 2

Show that the $4 \times 4$ forward DCT matrix given in Eq. (5.20) is unitary.

Victor Salazar

Numerade Educator

01:34

Problem 3

Show that the the $4 \times 4$ forward DWT (Haar) transformation matrix given in Example5.9 is unitary.

Victor Salazar

Numerade Educator

Calculate (by hand or using calculator) the DFT, DCT and Haar coefficients of the signal $f(n)=[1,1,0,0]$. Calculate the total energy in the signal domain, and in the frequency (i.e., DFT, DCT, and Haar) domain. Show that the total energies in all three transform domains are individually identical to the total signal energy.

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00:50

Problem 5

Calculate the DCT of the sequence $f(n)=[0.5,-0.5,-0.5,0.5]$. You will find that only one DCT coefficient is non-zero. Can you explain why is it so?

Michael Cryderman

Numerade Educator

Problem 6

Using MATLAB, calculate the DFT coefficients of the following signal
$$
f(n)= \begin{cases}1 & 0 \leq n \leq 4 \\ 0 & 5 \leq n \leq 15\end{cases}
$$
Plot the amplitude spectrum (in this particular case you can verify that the spectrum follows a sinc pattern). Show that the DFT preserves the signal energy.

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04:56

Problem 7

Repeat the above problem for the DCT.

Zulfiqar Ali

Numerade Educator

01:50

Problem 8

Repeat the above problem for 2-stage Daub-4 wavelet.

Carlos Henrique De Lima

Numerade Educator

05:07

Problem 9

Calculate the total energy of the 4 largest (absolute value) coefficients in each of the above three cases. Compare the energy compaction performance of the three transforms: DFT, DCT, and DWT in this particular case.

Dr. Satish Ingale

Numerade Educator

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Problem 10

Can we consider the block transforms as a special case of filterbank?

Rashmi Sinha

Numerade Educator

00:28

Problem 11

Compare and contrast orthogonal transform and subband decomposition.

Morgan Thompson

Numerade Educator

01:59

Problem 12

Show that the Daub-4 wavelet filter coefficients satisfy Eq. (5.24).

Suzanne W.

Numerade Educator

03:06

Problem 13

Write down $8 \times 8$ forward transform matrix for Haar wavelet. Show that the matrix is unitary.

Jack Chen

Numerade Educator

Problem 14

Using separable approach, calculate the DFT and DCT coefficients of the following $3 \times 3$ image.
$$
\left[\begin{array}{lll}
4 & 9 & 5 \\
2 & 7 & 3 \\
5 & 6 & 2
\end{array}\right]
$$
Show that the energy is preserved in the transform domain.

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00:19