Digital Signal Processing Using MATLAB and Wavelets
By Michael Weeks

Chapter 9.13 - Review Questions

  1. Find (by hand) what the signals $z_d[n]$ and $w_d[n]$ would be for the filter bank in Figure 9.11. Let $x = \{8, 4, 0, 6, 3, 7, 2, 9\}$, $a = \frac{1}{2}$, and $b = \frac{1}{2}$. Be sure to show your work.
  2. With a two-channel filter bank, such as in Figure 9.12, we found that the output

    $y[n] = (ac + bd) x[n-1] + (aa + bb + cc + dd) x[n-3] + (ac + bd) x[n-5]$.

    The Daubechies wavelet transform uses coefficients

    $a = \frac{1 - \sqrt{3}}{4 \sqrt{2}}$, $b = \frac{3 - \sqrt{3}}{4 \sqrt{2}}$, $c = \frac{3 + \sqrt{3}}{4 \sqrt{2}}$, and $d = \frac{1 + \sqrt{3}}{4 \sqrt{2}} .$

    a. What 2 constraints do the previous equation put on the coefficients? Show that the Daubechies coefficients satisfy these constraints.

    b. What effect do the down/up-samplers have on the output $y[n]$? How would removing them change the previous $y[n]$ equation?
  3. What is multiresolution (i.e., a wavelet transform having more than one octave)? Demonstrate this idea with a figure.

  4. For the filter coefficients below, use MATLAB to plot the frequency magnitude response, and the phase response. Determine if the filters have linear phase, and determine if the filters are highpass, lowpass, bandpass, or bandstop.

    a. {$-0.0884$, 0.0884, 0.7071, 0.7071, 0.0884, $-0.0884$}
    b. {$-0.1294$, 0.2241, 0.8365, 0.4830}
    c. {$-0.4830$, 0.8365, $-0.2241$, $-0.1294$}
  5. Write a MATLAB program to perform the following transform (for one octave) and inverse transform. See Figure 9.28, where LPF = {1, 1}, HPF = {$-1$, 1}. (LPF stands for Low Pass Filter, and HPF stands for High Pass Filter).

    Figure 9.28: Analysis filters.
    .

For the inverse transform, see Figure 9.29.

Figure 9.29: Synthesis filters.

Where ILPF = {$-1$, $-1$}, IHPF = {$-1$, 1} (ILPF stands for Inverse Low Pass Filter, and IHPF stands for Inverse High Pass Filter). Is $y$ a scaled version of $x$, e.g., $y[n] \times (-1/4) = x[n]$?

  • Implement a Haar transform, with MATLAB commands, for three octaves.

  • We saw a pattern for a CQF, in that the coefficients of one filter are alternately negated and mirrored, so that if we know one filter's coefficients, we can find the coefficients for the three others. Does this work in general? Would this work for $a = 3$, $b = 1$ in Figure 9.4?

  • If the down-samplers keep only one of every four values, what effect would this have on the filter bank? What if the filter bank structure were modified to have four channels?

  • Write a function to return the one octave, Daubechies four-coefficient wavelet transform for a given signal. Include low- and highpass outputs.

  • Given the input signal $x[n] = 2 \cos(2 \pi 100 n / 300) + \cos(2 \pi 110 n / 300 - \pi/8)$ for $n = 0..255$, write the commands to find the DWT for 3 octaves. Compare your results with those of the dwt function. Plot the original function, as well as the approximate signals.

  • For the Haar transform show in Figure 9.4, use values $a = b = \frac{1}{2}$ (no down-sampling), find signals $z$, $w$, and $y$, given an input of $x$ = {6, 1, 3, 7, 2, 5, 8, 10}.
  • Suppose you have a 3-octave DWT. Draw the analysis structure in terms of filters and down-samplers.

  • For a four-octave DWT, suppose the input has 1024 samples. How long would the detail outputs be? How long would the approximate outputs be? What if down/up-sampling were not used? For simplicity, you can assume that the filter's outputs are the same lengths as their inputs.

  • For an input of 1024 samples, how many octaves of the DWT could we have before the approximation becomes a single number? For simplicity, you can assume that the filter's outputs are the same lengths as their inputs.

  • We stated above that the quadrature mirror filter will not work for four coefficients, i.e., $h_0 = g_0 = \{a, b, c, d\}$, $h_1 = \{a, -b, c, -d\}$, $g_1 = -h_1$. Show analytically that this will not work. That is, what constraints are there on the values $a$, $b$, $c$, and $d$, in order to get perfect reconstruction?
    • © 2026 Infinity Science Press
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