$$

← Back to LINEAR SYSTEMS AND SIGNALS Overview

$$[2^{n-1} - (-2)^{n-1}]u[n]$$

(l) $\frac{(\ln \alpha)^n}{n!}u[n]$

• 5.1-5 Showing all work, evaluate $^{∞ n}=^{0 n}(−3/2)−n.
• 5.1-6 Using only the z-transforms of Table 5.1, determine the z-transform of each of the following signals.

(a) u[n] −u[n−2] (b) ^{γ n}−2u[n^{− 2}] (c) 2n+1u[n^{− 1}] +en−1u[n] (d) ²−ⁿ cos^π 3 n u[n−1] (e) ^nγ nu[n^{− 1}] (f) ⁿ(n−1)(n−2)2n−3u[n−m] for m = 0, 1, 2, 3 (g) (−1)nnu[n]

(h)

$$\sum_{k=0}^{\infty} k\delta(n-2k+1)$$

5.1-7 Find the inverse unilateral z-transform of each of the following: (a) ^z(^{z −}4)

(a)

$$z^2 - 5z + 6$$

\n(b) $\frac{z-4}{z^2 - 5z + 6}$
\n(c) $\frac{(e^{-2} - 2)z}{(z - e^{-2})(z - 2)}$
\n(d) $\frac{(z - 1)^2}{z^3}$
\n(e) $\frac{z(2z + 3)}{(z - 1)(z^2 - 5z + 6)}$
\n(f) $\frac{z(-5z + 22)}{(z + 1)(z - 2)^2}$
\n(g) $\frac{z(1.4z + 0.08)}{(z - 0.2)(z - 0.8)^2}$

(h)

$$\frac{z(z-2)}{z^2 - z + 1}$$

\n(i)

$$\frac{2z^2 - 0.3z + 0.25}{z^2 + 0.6z + 0.25}$$

\n(j)

$$\frac{2z(3z-23)}{(z-1)(z^2 - 6z + 25)}$$

\n(k)

$$\frac{z(3.83z + 11.34)}{(z-2)(z^2 - 5z + 25)}$$

\n(k)

$$\frac{z^2(-2z^2 + 8z - 7)}{(z^2 - 2z^2 + 8z - 7)}$$ $$(1) \frac{z^2(-2z^2+8z-7)}{(z-1)(z-2)^3}$$

5.1-8 (a) Expanding X[z] as a power series in z−1, find the first three terms of x[n] if

$$X[z] = \frac{2z^3 + 13z^2 + z}{z^3 + 7z^2 + 2z + 1}$$

(b) Extend the procedure used in part (a) to find the first four terms of x[n] if

$$X[z] = \frac{2z^4 + 16z^3 + 17z^2 + 3z}{z^3 + 7z^2 + 2z + 1}$$

• 5.1-9 A right-sided signal x[n] has z-transform given by ^X[z] = ^z6+2z5+3z4+4z³ *^z*4−¹ . Using a power series expansion of X[z], determine x[n] over −5 ≤ n ≤ 5.
• 5.1-10 Find x[n] by expanding

$$X[z] = \frac{\gamma z}{(z - \gamma)^2}$$

as a power series in z−1.

• 5.1-11 (a) In Table 5.1, if the numerator and the denominator powers of X[z] are M and N, respectively, explain why in some cases N − M = 0, while in others N − M = 1 or N − M = m (m any positive integer).
  • (b) Without actually finding the z-transform, state what is N − M for X[z] corresponding to ^x[n] = ^γ nu[n−4].
• 5.2-1 For a discrete-time signal shown in Fig. P5.2-1, show that

$$X[z] = \frac{1 - z^{-m}}{1 - z^{-1}}$$

Find your answer by using the definition in Eq. (5.1) and by using Table 5.1 and an appropriate property of the z-transform.

Figure P5.2-1

• 5.2-2 Determine the unilateral z-transform of signal ^x[n] = (1−n) cos ^{π 2} (n−1) u[n−1].
• 5.2-3 Suppose a DT signal x[n] = 2(u[n−10] −u[n− ⁶]) has a transform ^X(z). Define ^Y(z) ^{= 1} 2z−3 d dzX(2z). Using graphic plot or vector notation, determine the corresponding signal y[n].
• 5.2-4 Suppose a DT signal x[n] = 3(u[n] −u[n−5]) has a transform X(z). Define Y(z) = 2z−^{4 d} dzX z 2 . Using graphic plot or vector notation, determine the corresponding signal y[n].
• 5.2-5 Find the z-transform of the signal illustrated in Fig. P5.2-5. Solve this problem in two ways, as in Exs. 5.2d and 5.4. Verify that the two answers are equivalent.
• 5.2-6 Using z-transform techniques and properties (no time-domain convolution sum!), determine the convolution ^y[n] =( ^{1 2} )nu[n−3]∗( ^{1 3} )n−6u[n−4]. Express your answer in the form y[n] = c1γ ⁿ−N^{1 1 u}[^{n −} *^N*1] + *^c*2^{γ n}−N^{2 2} u[n − N2], making sure to clearly identify the constants c1, c2, γ1, γ2, N1, and N2.

Figure P5.2-5

5.2-7 Determine the inverse unilateral z-transform x[n] of the signal

$$X[z] = \frac{d^7}{dz^7} \left[ \frac{z^{-4}}{(z - \frac{1}{2})(z + 3)} \right]$$

• 5.2-8 Using only the fact that ^γ nu[n]⇐⇒z/(z−γ ) and properties of the z-transform, find the z-transform of each of the following:
  • (a) n2u[n]
  • (b) *ⁿ*2^γ nu[n]
  • (c) n3u[n]
  • (d) an[u[n] −u[n−m]]
  • (e) ne−²nu[n−m]
  • (f) (n−2)(0.5)n−^{3 u}[n−4]
• 5.2-9 Using only pair 1 in Table 5.1 and appropriate properties of the z-transform, derive iteratively pairs 2 through 9. In other words, first derive pair 2. Then use pair 2 (and pair 1, if needed) to derive pair 3, and so on.
• 5.2-10 Find the z-transform of cos(πn/4)u[n] using only pairs 1 and 11b in Table 5.1 and a suitable property of the z-transform.
• 5.2-11 Apply the time-reversal property to pair 6 of Table 5.1 to show that ^γ nu[−(^{n + 1})] ⇐⇒ −z/(z−γ ) and the ROC is given by |z| < |γ |.
• 5.2-12 (a) If x[n] ⇐⇒ X[z], then show that (−1)nx[n] ⇐⇒ ^X[−z].
  • (b) Use this result to show that (−γ )nu[n] ⇐⇒ z/(z +γ ).
  • (c) Use these results to find the z-transforms of ^xi[n]=[2n−^{1 −}(−2)n−1]u[n] and ^x−ii[n] = ^{γ n} cosπn u[n]
• 5.2-13 (a) If x[n] ⇐⇒ X[z], then show that

$$\sum_{k=0}^{n} x[k] \Longleftrightarrow \frac{zX[z]}{z-1}$$

578 CHAPTER 5 DISCRETE-TIME SYSTEM ANALYSIS USING THE Z-TRANSFORM

• (b) Use this result to derive pair 2 from pair 1 in Table 5.1.
• 5.2-14 A number of causal time-domain functions are shown in Fig. P5.2-14. List the function of time that corresponds to each of the following functions of z. Few or no calculations are necessary! Be careful, the graphs may be scaled differently.

(a)

$$\frac{z^2}{(z-0.75)^2}$$

\n(b) $\frac{z^2 - 0.9z/\sqrt{2}}{z^2 - 0.9\sqrt{2}z + 0.81}$
\n(c) $\sum_{k=1}^{4} z^{-2k}$

$$k=0$$

\n(d)

\n

$$\frac{z^{-5}}{1-z^{-1}}$$

(e)

$$\frac{z^2}{z^4 - 1}$$

(c) 0.75z

(f)

$$\frac{1}{(z-0.75)^2}$$

\n(g) $\frac{z^2 - z/\sqrt{2}}{z-\sqrt{2}}$

(g)

$$\frac{z^2 - \sqrt{2}z + 1}{z^2 - 5z^{-5} + 4z^{-6}}$$

\n(h)

$$\frac{z^{-1} - 5z^{-5} + 4z^{-6}}{5(1 - z^{-1})^2}$$

\n(i)

$$\frac{z}{z - 1.1}$$

(j)

$$\frac{0.25z^{-1}}{(1-z^{-1})(1-0.75z^{-1})}$$

• 5.2-15 Suppose we upsample a causal signal x[n] by factor N to produce signal y[n]. Express Y(z) in terms of X(z), taking care to mathematically justify your result.
• 5.3-1 Using z-transform techniques, find the output y[n] of an LTID system specified by the equation ^y[n]− ^{1 3} y[n−1] = x[n−1] if the initial condition is y[−1] = 2 and the input is x[n]=−u[n].
• 5.3-2 Consider an LTID system y[n] − y[n − 2] = x[n] with y[−1] = 0, y[−2] = 1, and x[n] = u[n].
  • (a) Determine Y[z], expressed as a rational function in standard factored form.
  • (b) Use Y[z] from part (a) to solve for the system output y[n].
• 5.3-3 Solve Prob. 3.8-23 by the z-transform method.
• 5.3-4 Consider a DT system with transfer function ^H[z] = ²z−^{2 z}−0.5 . Assuming the system is both controllable and observable, determine the ZIR yzir[n] given y[−1] = 1.
• 5.3-5 (a) Solve

$$y[n+1] + 2y[n] = x[n+1]$$

when

$$y[0] = 1$$

and $x[n] = e^{-(n-1)}u[n]$

• (b) Find the zero-input and the zero-state components of the response.

• 5.3-6 Consider a LTID system that is described by the difference equation ^y[n] − ^{1 4} y[n − 2] = x[n−1].

• (a) Use transform-domain techniques to determine the zero-state response yzsr[n] to input x[n] = 3u[n−5].

• (b) Use transform-domain techniques to determine the zero-input response yzir[n] given yzir[−2] = yzir[−1] = 1.

• 5.3-7 (a) Find the output y[n] of an LTID system specified by the equation

$$2y[n+2] - 3y[n+1] + y[n]$$

= $4x[n+2] - 3x[n+1]$

for input ^x[n] = (4)−nu[n] and initial conditions y[−1] = 0 and y[−2] = 1.

• (b) Find the zero-input and the zero-state components of the response.
• (c) Find the transient and the steady-state components of the response.
• 5.3-8 Solve Prob. 5.3-7 if initial conditions y[−1] and y[−2] are instead replaced with auxiliary conditions y[0] = 3/2 and y[1] = 35/4.
• 5.3-9 (a) Solve

$$4y[n+2] + 4y[n+1] + y[n] = x[n+1]$$

with y[−1] = 0, y[−2] = 1, and x[n] = u[n].

• (b) Find the zero-input and the zero-state components of the response.
• (c) Find the transient and the steady-state components of the response.
• 5.3-10 Solve

$$y[n+2] - 3y[n+1] + 2y[n] = x[n+1]$$

if ^y[−1] = 2, ^y[−2] = 3, and ^x[n] = (3)nu[n].

5.3-11 Solve

$$y[n+2] - 2y[n+1] + 2y[n] = x[n]$$

with y[−1] = 1, y[−2] = 0, and x[n] = u[n].

• 5.3-12 Consider a causal LTID system described as ^H(z) ^{= 21}(z2+1) ¹⁶(z2^{+ 1 4 z− 3} 8 ) .

  • (a) Determine the standard delay-form difference equation description of this system.
  • (b) Using transform-domain techniques, determine the system impulse response h[n].
  • (c) Using transform-domain techniques, determine yzir[n] given y[−1] = 16 and y[−2] = 8.

• 5.3-13 Consider a causal LTID system described as ^y[n] − ^{5 6 y}[^{n − 1}] + ^{1 6 y}[^{n − 2}] = ^{3 2 x}[^{n − 1}] + ^{3 2} x[n−2].

  • (a) Determine the (standard-form) system transfer function H(z) and sketch the system pole-zero plot.
  • (b) Using transform-domain techniques, determine yzir[n] given y[−1] = 2 and y[−2]=−2.

• 5.3-14 Solve

$$y[n] + 2y[n-1] + 2y[n-2]$$

= $x[n-1] + 2x[n-2]$

with ^y[0] = 0, ^y[1] = 1, and ^x[n] = enu[n].

• 5.3-15 A system with impulse response h[n] = ²(1/3)nu[^{n − 1}] produces an output ^y[n] = (−2)nu[^{n − 1}]. Determine the corresponding input x[n].
• 5.3-16 A professor recently received an unexpected $10 (a futile bribe attached to a test). Being the savvy investor that she is, the professor decides to invest the $10 into a savings account that earns 0.5% interest compounded monthly (6.17% APY). Furthermore, she decides to supplement this initial investment with an additional $5 deposit made every month, beginning the month immediately following her initial investment.
  • (a) Model the professor’s savings account as a constant coefficient linear difference equation. Designate y[n] as the account balance at month n, where n = 0 corresponds to the first month that interest is awarded (and that her $5 deposits begin).
  • (b) Determine a closed-form solution for y[n]. That is, you should express y[n] as a function only of n.
  • (c) If we consider the professor’s bank account as a system, what is the system impulse response h[n]? What is the system transfer function H[z]?
  • (d) Explain this fact: if the input to the professor’s bank account is the everlasting exponential ^x[n] = ^{1n =} 1, then the output is not ^y[n] = ¹nH[1] = ^H[1].
• 5.3-17 Sally deposits $100 into her savings account on the first day of every month except for each December, when she uses her money to buy

holiday gifts. Define b[m] as the balance in Sally’s account on the first day of month m. Assume Sally opens her account in January (m = 0), continues making monthly payments forever (except each December!), and that her monthly interest rate is 1%. Sally’s account balance satisfies a simple difference equation b[m] = (1.01)b[m − 1] + p[m], where p[m] designates Sally’s monthly deposits. Determine a closed-form expression for b[m] that is only a function of the month m.

• 5.3-18 For each impulse response, determine the number of system poles, whether the poles are real or complex, and whether the system is BIBO-stable.
  • (a) *^h*1[n] = (−1+(0.5)n)u[n]
  • (b) *^h*2[n] = (j)n(u[n] −u[n−10])
• 5.3-19 Find the following sums:

(a)

$$\sum_{k=0}^{n} k$$

(b)

$$\sum_{k=0}^{n} k^2$$

k=0 [Hint: Consider a system whose output y[n] is the desired sum. Examine the relationship between y[n] and y[n − 1]. Note also that y[0] = 0.]

5.3-20 Find the following sum:

$$\sum_{k=0}^{n} k^3$$

[Hint: See the hint for Prob. 5.3-19.]

5.3-21 Find the following sum:

$$\sum_{k=0}^{n} ka^k \qquad a \neq 1$$

[Hint: See the hint for Prob. 5.3-19.]

• 5.3-22 Redo Prob. 5.3-19 using the result in Prob. 5.2-13a.
• 5.3-23 Redo Prob. 5.3-20 using the result in Prob. 5.2-13a.
• 5.3-24 Redo Prob. 5.3-21 using the result in Prob. 5.2-13a.

5.3-25 (a) Find the zero-state response of an LTID system with transfer function

$$H[z] = \frac{z}{(z+0.2)(z-0.8)}$$

and the input ^x[n] = ^e(n+1) u[n].

(b) Write the difference equation relating the output y[n] to input x[n].

5.3-26 Repeat Prob. 5.3-25 for x[n] = u[n] and

$$H[z] = \frac{2z+3}{(z-2)(z-3)}$$

5.3-27 Repeat Prob. 5.3-25 for

$$H[z] = \frac{6(5z - 1)}{6z^2 - 5z + 1}$$

• and the input x[n] is (a) (4)−nu[n] (b) (4)−(n−2) u[n−2] (c) (4)−(n−2) u[n] (d) (4)−nu[n−2]
• 5.3-28 Repeat Prob. 5.3-25 for x[n] = u[n] and

$$H[z] = \frac{2z - 1}{z^2 - 1.6z + 0.8}$$

• 5.3-29 Find the transfer functions corresponding to each of the systems specified by difference equations in Probs. 5.3-5, 5.3-7, 5.3-9, and 5.3-14.
• 5.3-30 Find h[n], the unit impulse response of the systems described by the following equations:
  • (a) y[n] +3y[n−1] +2y[n−2] = x[n] +3x[n− 1] +3x[n−2]
  • (b) y[n + 2] + 2y[n + 1] + y[n] = 2x[n + 2] − x[n+1]
  • (c) y[n]−y[n−1]+0.5y[n−2] = x[n]+2x[n− 1]
• 5.3-31 Find h[n], the unit impulse response of the systems in Probs. 5.3-25, 5.3-26, and 5.3-28.
• 5.3-32 A system has impulse response h[n] = u[n − 3].
  • (a) Determine the impulse response of the inverse system h−1[n].
  • (b) Is the inverse stable? Is the inverse causal?
  • (c) Your boss asks you to implement h−1[n] to the best of your ability. Describe your realizable design, taking care to identify any deficiencies.

5.4-1 A system has impulse response given by

$$h[n] = \left[ \left( \frac{1+j}{\sqrt{8}} \right)^n + \left( \frac{1-j}{\sqrt{8}} \right)^n \right] u[n]$$

This system can be implemented according to Fig. P5.4-1.

Figure P5.4-1

• (a) Determine the coefficients A¹ and A² to implement h[n] using the structure shown in Fig. P5.4-1.
• (b) What is the zero-state response y0[n] of this system, given a shifted unit step input x[n] = u[n+3]?
• 5.4-2 (a) Show the canonic direct form, a cascade, and a parallel realization of

$$H[z] = \frac{z(3z - 1.8)}{z^2 - z + 0.16}$$

• (b) Find the transpose of the realizations obtained in part (a).
• 5.4-3 Repeat Prob. 5.4-2 for

$$H[z] = \frac{5z + 2.2}{z^2 + z + 0.16}$$

5.4-4 Repeat Prob. 5.4-2 for

$$H[z] = \frac{3.8z - 1.1}{(z - 0.2)(z^2 - 0.6z + 0.25)}$$

5.4-5 Repeat Prob. 5.4-2 for

$$H[z] = \frac{z(1.6z - 1.8)}{(z - 0.2)(z^2 + z + 0.5)}$$

5.4-6 Repeat Prob. 5.4-2 for

$$H[z] = \frac{z(2z^2 + 1.3z + 0.96)}{(z + 0.5)(z - 0.4)^2}$$

5.4-7 Consider the LTID system shown in Fig. P5.4-7.

$$x[n]$$

$\rightarrow$ $z^{-1}$ $\rightarrow$ $z^{-1}$ $\rightarrow$ $z^{-1}$ $\rightarrow$ $y[n]$

• Figure P5.4-7
• (a) Determine the standard delay-form difference equation description of this system.
• (b) Determine the impulse response h[n] of this system.
• (c) Is this realization canonical? Explain.
• (d) Is this system stable? Explain.
• (e) Is this system causal? Explain.
• 5.4-8 Realize a system whose transfer function is

$$H[z] = \frac{2z^4 + z^3 + 0.8z^2 + 2z + 8}{z^4}$$

5.4-9 Realize a system whose transfer function is given by

$$H[z] = \sum_{n=0}^{6} nz^{-n}$$

5.4-10 Consider the LTID system shown in Fig. P5.4-10, where parameter c is an arbitrary, real constant.

Figure P5.4-10

• (a) Determine the system transfer function H[z], expressed in standard rational form.
• (b) Determine all system poles and all system zeros.
• (c) Is the system of Fig. P5.4-10 canonical? Explain.
• (d) What constraints, if any, exist on parameter c to ensure that the system is stable?
• 5.4-11 This problem demonstrates the enormous number of ways of implementing even a relatively low-order transfer function. A second-order transfer function has two real zeros and two real poles. Discuss various ways of realizing

such a transfer function. Consider canonic direct, cascade, parallel, and the corresponding transposed forms. Note also that interchange of cascaded sections yields a different realization.

5.4-12 Consider a digital audio system: an input analog-to-digital converter (ADC) is used to collect input samples at a CD-quality rate Fs = 44 kHz. Input samples are processed with a digital filter to generate output samples, which are sent to a digital-to-analog converter (DAC) at the same rate Fs. Every sample interval T = 1/Fs, the digital processor executes the following MATLAB-compatible code:

% Read input sample from the ADC
x = read_ADC;
% Process input and...
mem(1) = x - mem(3)*9/16;
% ...compute output sample
y = mem(1)*7/16 - mem(3)*7/16;
% Send output sample to the DAC
write_DAC = y;
% Update memory for next iteration
mem(3) = mem(2);
mem(2) = mem(1);

• (a) Does the code implement DFI, DFII, TDFI, or TDFII? Support your answer by drawing the appropriate block diagram labeled in a manner that is consistent with the code.
• (b) Determine the transfer function H[z] of this system.
• (c) What is the basic filtering function of this system: LP, HP, BP, or BS? Justify your answer.
• (d) Determine the transfer function H−1[z] of the inverse system to H[z] and draw its DFI block implementation. How well will the inverse system operate?
• 5.4-13 Repeat Prob. 5.4-12 but instead use the code:

% Read input sample from the ADC
x = read_ADC;
% Compute output sample
y = x*7/32+mem(1);
% Send output sample to the DAC
write_DAC = y;
% Update memory for next iteration
mem(1) = mem(2);
mem(2) = x*7/32 + y*9/16;

• 5.5-1 A CT sinusoid x(t) = cos(ωt) is sampled at a greater-than-Nyquist rate F^s = 1000 Hz to produce a DT sinusoid x(t) = cos(n). Determine the analog frequency ω if (a) ^{= π} 4
  • (b) ^{= 2π} 3
  • (c) ^{= 7} 8
• 5.5-2 Find the amplitude and phase response of the digital filters depicted in Fig. P5.5-2.
• 5.5-3 A causal LTID system ^H(z) ^{= 21}(z−j)(z+j) ¹⁶(z^{− 1 2} )(z^{+ 3} 4 ) has a periodic input x[n] that toggles between the values 1 and 2. That is, x[n]=[…, 1, 2, 1, ↓ 2 , 1, 2, 1, …], where x[0] = 2.
  • (a) Plot the magnitude response ^|H(ej)^| over −2π ≤ ≤ 2π.
  • (b) Plot the phase response H(ej) over −2π ≤ ≤ 2π.
  • (c) Determine the system output y[n] in response to the periodic input x[n].

5.5-4 A causal LTID system ^H(z) ^{= −}7(z+1) ³²(z−^{j 3 4} )(z+^{j 3} 4 ) has a periodic input x[n] that cycles through the 4 values 3, 2, 1, and 2. That is, x[n] = […, 3, 2, 1, 2, ↓ 3, 2, 1, 2, …], where x[0] = 3.

• (a) Plot the magnitude response ^|H(ej)^| over −2π ≤ ≤ 2π.
• (b) Plot the phase response H(ej) over −2π ≤ ≤ 2π.
• (c) Determine the system output y[n] in response to the periodic input x[n].
• 5.5-5 Find the amplitude and the phase response of the filters shown in Fig. P5.5-5. [Hint: Express ^H[ej] as ^e−j2.5Ha[ej].]
• 5.5-6 Find the frequency response for the moving-average system in Prob. 3.4-3. The input–output equation of this system is given by

$$y[n] = \frac{1}{5} \sum_{k=0}^{4} x[n-k]$$

5.5-7 (a) Input–output relationships of two filters are described by

(i) y[n]=−0.9y[n−1] +x[n]

(ii) y[n] = 0.9y[n−1] +x[n]

For each case, find the transfer function, the amplitude response, and the phase response. Sketch the amplitude response, and state the type (highpass, lowpass, etc.) of each filter.

• (b) Find the response of each of these filters to a sinusoid x[n] = cosn for = 0.01π and 0.99π. In general, show that the gain (amplitude response) of filter (i) at frequency ⁰ is the same as the gain of filter (ii) at frequency π −0.
• 5.5-8 For an LTID system specified by the equation

$$y[n+1] - 0.5y[n] = x[n+1] + 0.8x[n]$$

(a) Find the amplitude and the phase response.

• (b) Find the system response y[n] for the input x[n] = cos(0.5k −(π/3)).
• 5.5-9 For an asymptotically stable LTID system, show that the steady-state response to input ejnu[n] is ^H[ej]ejnu[n]. The steady-state response is that part of the response which does not decay with time and persists forever.
• 5.5-10 Express the following signals in terms of apparent frequencies:
  • (a) cos(0.8πn+θ )
  • (b) sin(1.2πn+θ )
  • (c) cos(6.9n+θ )

Figure P5.5-2

(b)

Figure P5.5-5

• (d) cos(2.8πn+θ ) +2 sin(3.7πn+θ )
• (e) sinc(πn/2)
• (f) sinc(3πn/2)
• (g) sinc(2πn)
• 5.5-11 Show that cos √ (0.6πn + (π/6)) + 3 cos(1.4πn + (π/3)) = 2 cos(0.6πn − (π/6)).
• 5.5-12 (a) A digital filter has the sampling interval T = 50µs. Determine the highest frequency that can be processed by this filter without aliasing.
  • (b) If the highest frequency to be processed is 50 kHz, determine the minimum value of the sampling frequency F^s and the maximum value of the sampling interval T that can be used.
• 5.5-13 Consider the discrete-time system represented by

$$y[n] = \sum_{k=0}^{\infty} (0.5)^k x[n-k]$$

• (a) Determine and plot the magnitude response ^|H[ej]| of the system.
• (b) Determine and plot the phase response ^H[ej] of the system.
• (c) Find an efficient block representation that implements this system.
• 5.6-1 Pole-zero configurations of certain filters are shown in Fig. P5.6-1. Sketch roughly the amplitude response of these filters.
• 5.6-2 Figure P5.6-2 displays the pole-zero plot of a second-order real, causal LTID system that has H[−1]=−1.

Figure P5.6-2

• (a) Determine the five constants b0, b1, b2, a1, and a² that specify the transfer function ^H[z] = ^b0z2+b1z+b^{2 z}2+a1z+a² .
• (b) Using the techniques of Sec. 5.6, accurately hand-sketch the system magnitude response ^|H[ej]| over the range (−2^{π ≤ ≤ 0}).
• (c) Determine the output y[n] of this system if the input is x[n] = sin ^πn 2 .
• 5.6-3 Repeat Prob. 5.6-2 if the zero at z = 1 is moved to z = −1 and H[1]=−1 is specified rather than H[−1]=−1.
• 5.6-4 Figure P5.6-4 displays the pole-zero plot of a second-order real, causal LTID system that has H[−1] = 1.
  • (a) Determine the five constants k, b1, b2, a1, and a² that specify the transfer function ^H[z] = ^{k z}2+b1z+b^{2 z}2+a1z+a² .

Figure P5.6-4

• (b) Using the techniques of Sec. 5.6, accurately hand-sketch the system magnitude response ^|H[ej]| over the range (−^{π ≤ ≤} π ).
• (c) A signal x(t) = cos(100πt) + cos(500πt) is sampled at a greater than Nyquist rate F^s Hz and then input into the above LTID system to produce DT output y[n] = β cos(0n + θ ). Determine F^s and 0. You do not need to find constants β and θ.
• (d) Is the inpulse response h[n] of this system absolutely summable? Justify your answer.
• 5.6-5 Figure P5.6-5 displays the pole-zero plot of a second-order real, causal LTID system that has H[1]=−1.

(a) Determine the five constants k, b1, b2, a1, and a² that specify the transfer function ^H[z] = ^{k z}2+b1z+b^{2 z}2+a1z+a² .

Figure P5.6-5

• (b) Using the techniques of Sec. 5.6, accurately hand-sketch the system magnitude response ^|H[ej]| over the range (−^{π ≤ ≤} π ).
• (c) A signal x(t) = cos(2πft) is sampled at a rate F^s = 1 kHz and then input into the above LTID system to produce DT output y[n]. Determine, if possible, the frequency or frequencies f that will produce zero output, y[n] = 0.
• 5.6-6 The system y[n] − y[n − 1] = x[n] − x[n − 1] is an all-pass system that has zero phase response. Is there any difference between this system and the system y[n] = x[n]? Justify your answer.
• 5.6-7 Figure P5.6-7 displays the pole-zero plot of a second-order real, causal LTID system that has a repeated zero and H[1] = 4. The solid circle is the unit circle.

Figure P5.6-7

• (a) Determine the five constants k, b1, b2, a1, and a² that specify the transfer function ^H[z] = ^{k z}2+b1z+b^{2 z}2+a1z+a² .
• (b) Using the techniques of Sec. 5.6, accurately hand-sketch the system magnitude response ^|H[ej]| over the range (−^{π ≤ ≤} π ).
• (c) Determine the steady-state output yss[n] of this system if the input is x[n] = cos( ³π*ⁿ* ⁴ )u[n].
• (d) State whether this system is LP, HP, BP, BS, or other. If the digital system operates at F^s = 8 kHz, what is the approximate hertzian cutoff frequency (or frequencies) of this system?
• 5.6-8 The magnitude and phase responses of a real, stable, LTI system are shown in Fig. P5.6-8.
  • (a) What type of system is this: lowpass, highpass, bandpass, or bandstop?
  • (b) What is the output of this system in response to

$$x_1[n] = 2\sin\left(\frac{\pi}{2}n + \frac{\pi}{4}\right)$$

(c) What is the output of this system in response to

$$x_2[n] = \cos\left(\frac{7\pi}{4}n\right)$$

• 5.6-9 Consider an LTID system with system function ^H[z] = ^b0 *^z*2+¹ *^z*2−9/¹⁶ .

  • (a) Determine the constant b⁰ so that the system frequency response at = −π is −1.
  • (b) Accurately sketch the system poles and zeros.
  • (c) Using the locations of the system poles and zeros, sketch ^|H[ej]| over 0 ^{≤ ≤ 2}π.

• (d) Determine the response y[n] to the input x[n] = (−1+j)+j ^{n +}(1−j)sin(πn+1).

• (e) Draw an appropriate block diagram representation of this system.

• 5.6-10 Do Prob. 5.10-3 by graphical procedure. Do the sketches approximately, without using MATLAB.

• 5.6-11 Do Prob. 5.10-8 by graphical procedure. Do the sketches approximately, without using MATLAB.

• 5.6-12 (a) Realize a digital filter whose transfer function is given by

$$H[z] = K \frac{z+1}{z-a}$$

• (b) Sketch the amplitude response of this filter, assuming |a| < 1.
• (c) The amplitude response of this lowpass filter is maximum at = 0. The 3 dB bandwidth is the frequency at which the amplitude response drops to 0.707 (or 1/ ^√2) times its maximum value. Determine the 3 dB bandwidth of this filter when a = 0.2.
• 5.6-13 Design a digital notch filter to reject frequency 5000 Hz completely and to have a sharp recovery on either side of 5000 Hz to a gain of unity. The highest frequency to be processed is 20 kHz (F^h = 20,000). [Hint: See Ex. 5.15. The zeros should be at e±jω*^T* for ω corresponding to 5000 Hz, and the poles are at ae±jω*^T* with a < 1. Leave your answer in terms of a. Realize this filter using the canonical form. Find the amplitude response of the filter.]

5.6-14 Consider the desired DT system magnitude response ^|H[ej]| in Fig. P5.6-14.

Figure P5.6-14

• (a) Is the filter LP, HP, BP, BS, or other? Explain.
• (b) Sketch the pole-zero plot of a 2nd-order system that behaves as a reasonable approximation of Fig. P5.6-14. What is the coefficient b0?
• 5.6-15 Show that a first-order LTID system with a pole at z = r and a zero at z = 1/r (r ≤ 1) is an allpass filter. In other words, show that the amplitude response ^|H[ej]| of a system with the transfer function

$$H[z] = \frac{z - \frac{1}{r}}{z - r} \qquad r \le 1$$

is constant with frequency. This is a first-order allpass filter. [Hint: Show that the ratio of the distances of any point on the unit circle from the zero (at z = 1/r) and the pole (at z = r) is a constant 1/r.]

Generalize this result to show that an LTID system with two poles at ^{z =} re±j^θ and two zeros at ^{z =} (1/r)e±j^θ (^{r ≤ 1}) is an allpass filter. In other words, show that the amplitude response of a system with the transfer function

$$H[z] = \frac{\left(z - \frac{1}{r}e^{j\theta}\right)\left(z - \frac{1}{r}e^{-j\theta}\right)}{(z - re^{j\theta})(z - re^{-j\theta})}$$ $$= \frac{z^2 - \left(\frac{2}{r}\cos\theta\right)z + \frac{1}{r^2}}{z^2 - (2r\cos\theta)z + r^2} \qquad r \le 1$$

is constant with frequency.

5.6-16 (a) If h1[n] and h2[n], the impulse responses of two LTID systems are related by h2[n] = (−1)nh1[n], then show that

$$H_2[e^{j\Omega}] = H_1[e^{j(\Omega \pm \pi)}]$$

How is the frequency response spectrum *^H*2[ej] related to the *^H*1[ej]?

• (b) If H1[z] represents an ideal lowpass filter with cutoff frequency c, sketch *^H*2[ej]. What type of filter is *^H*2[ej]?
• 5.6-17 Mappings such as the bilinear transformation are useful in the conversion of continuous-time filters to discrete-time filters. Another useful type of transformation is one that converts a discrete-time filter into a different type of discrete-time filter. Consider a transformation that replaces z with −z.
  • (a) Show that this transformation converts lowpass filters into highpass filters and highpass filters into lowpass filters.
  • (b) If the original filter is an FIR filter with impulse response h[n], what is the impulse response of the transformed filter?
• 5.6-18 The bilinear transformation is defined by the rule s = 2(1−z−1)/T(1+z−1).
  • (a) Show that this transformation maps the ω axis in the ^s plane to the unit circle ^{z =} ej in the z plane.
  • (b) Show that this transformation maps to 2 arctan(ωT/2).
• 5.7-1 In Ch. 3, we used another approximation to find a digital system to realize an analog system. We showed that an analog system specified by Eq. (3.12) can be realized by using the digital system specified by Eq. (3.13). Compare that solution with the one resulting from the

impulse-invariance method. Show that one result is a close approximation of the other and that the approximation improves as T → 0.

• 5.7-2 A CT system has impulse response hct(t) = e−^t u(t). Draw the DFI realization of the corresponding DT system designed by the impulse-invariance method with T = 0.1.
• 5.7-3 (a) Using the impulse-invariance criterion, design a digital filter to realize an analog filter with transfer function

$$H_a(s) = \frac{7s + 20}{2(s^2 + 7s + 10)}$$

• (b) Show a canonical and a parallel realization of the filter. Use a 1% criterion for the choice of T.
• 5.7-4 Use the impulse-invariance criterion to design a digital filter to realize the second-order analog Butterworth filter with transfer function

$$H_a(s) = \frac{1}{s^2 + \sqrt{2}s + 1}$$

Use a 1% criterion for the choice of T.

• 5.7-5 Design a digital integrator using the impulse-invariance method. Find and give a rough sketch of the amplitude response, and compare it with that of the ideal integrator. If this integrator is used primarily for integrating audio signals (whose bandwidth is 20 kHz), determine a suitable value for T.

• 5.7-6 An oscillator by definition is a source (no input) that generates a sinusoid of a certain frequency ω0. Therefore, an oscillator is a system whose zero-input response is a sinusoid of the desired frequency. Find the transfer function of a digital oscillator to oscillate at 10 kHz by the methods described in parts (a) and (b). In both methods, select T so that there are 10 samples in each cycle of the sinusoid.

  • (a) Choose H[z] directly so that its zero-input response is a discrete-time sinusoid of frequency =ωT corresponding to 10 kHz.
  • (b) Choose Ha(s) whose zero-input response is an analog sinusoid of 10 kHz. Now use the impulse invariance method to determine H[z].
  • (c) Show a canonical realization of the oscillator.

• 5.7-7 A variant of the impulse invariance method is the step-invariance method of digital filter synthesis. In this method, for a given Ha(s), we design H[z] in Fig. 5.24a such that y(nT) in Fig. 5.24b is identical to y[n] in Fig. 5.24a when x(t) = u(t).

  • (a) Show that, in general,

$$H[z] = \frac{z-1}{z} \mathcal{Z} \left[ \left( \mathcal{L}^{-1} \frac{H_a(s)}{s} \right)_{t=kT} \right]$$

(b) Use this method to design H[z] for

$$H_a(s) = \frac{\omega_c}{s + \omega_c}$$

• (c) Use the step-invariance method to synthesize a discrete-time integrator and compare its amplitude response with that of the ideal integrator.
• 5.7-8 Use the ramp-invariance method to synthesize a discrete-time differentiator and integrator. In this method, for a given Ha(s), we design H[z] such that y(nT) in Fig. 5.24b is identical to y[n] in Fig. 5.24a when x(t) = tu(t).
• 5.7-9 In an impulse-invariance design, show that if Ha(s) is a transfer function of a stable system, the corresponding H[z] is also a transfer function of a stable system.
• 5.7-10 First-order backward differences provide the transformation rule s = (1−z−1)/T.
  • (a) Show that this transformation maps the ω axis in the s plane to a circle of radius 1/2 centered at (1/2, 0) in the z plane.
  • (b) Show that this transformation maps the left-half s plane to the interior of the unit circle in the z plane, which ensures that stability is preserved.
• 5.8-1 Find the z-transform (if it exists) and the corresponding ROC for each of the following signals:
  • (a) (0.8)nu[n] +2nu[−(n+1)]

(b)

$$2^n u[n] - 3^n u[-(n+1)]$$

(c)

$$(-2)^{n+3}u[-n] + \sum_{k=0}^{\infty} (0.5)^{k-1} \delta(n-2k)$$

(d)

$$(0.8)^n u[n] + (0.9)^n u[-(n+1)]$$

(e)

$$[(0.8)^n + 3(0.4)^n]u[-(n+1)]$$

(f) [(0.8)^{n +}3(0.4)n]u[n]

(g)

$$(0.8)^n u[n] + 3(0.4)^n u[-(n+1)]$$ $$(0.6) \quad u[n] + 5(0.4) \quad u[$$

(h)

$$(0.5)^{|n|}$$ $$(i) \t n u[-(n+1)]$$

• 5.8-2 Using the definition, compute the bilateral z-transform X(z) of (a) ^x[n] = ³nu[−n] (b) ^x[n] = ( ^{1 3} )nu[n] Express your answers in standard rational form.
• 5.8-3 Determine the inverse z-transform x[n] of ^X[z] = *^z*2^{− 1} 3 z (z−1)(z+2) with ROC 1 ^&lt; ^|z^{| &}lt; 2.

5.8-4 Find the inverse

$$z$$

-transform of

$$X[z] = \frac{(e^{-2} - 2)z}{(z - e^{-2})(z - 2)}$$

when the ROC is,
\n(a)

$$|z| > 2$$

\n(b) $e^{-2} < |z| < 2$
\n(c) $|z| < e^{-2}$

5.8-5 Use partial fraction expansions, z-transform tables, and a region of convergence (|z| < 1/2) to determine the inverse z-transform of

$$X(z) = \frac{1}{(2z+1)(z+1)(z+\frac{1}{2})}$$

• 5.8-6 Using z-transform techniques and properties (no time-domain convolution sum!), determine the convolution ^y[n] = ( ^{1 3} )n−3u[^{n − 2}] ∗ (2)nu[−n]. Express your answer in the form y[n] = c1γ ^{n 1 u}[^{n +} *^N*1] + *^c*2^{γ n 2} u[−n + N2], making sure to clearly identify the constants c1, c2, γ1, γ2, N1, and N2.
• 5.8-7 Using partial fraction expansions, z-transform tables, and the fact that h[n] is stable, determine the inverse z-transform of

$$H[z] = \frac{z^4 + z^3}{(z - 2)(z + \frac{1}{2})}.$$

5.8-8 Consider the system

$$H[z] = \frac{z(z - \frac{1}{2})}{(z^3 - \frac{27}{8})}$$

• (a) Draw the pole-zero diagram for H[z] and identify all possible regions of convergence.
• (b) Draw the pole-zero diagram for H−1[z] and identify all possible regions of convergence.
• 5.8-9 A discrete-time signal x[n] has a rational z-transform that contains a pole at z = 0.5. Given *^x*1[n] = (1/3)nx[n] is absolutely summable and

Problems 589

*^x*2[n] = (1/4)nx[n] is not absolutely summable, determine whether x[n] is left-sided, right-sided, or two-sided. Justify your answer!

• 5.8-10 Let x[n] be an absolutely summable signal with rational z-transform X[z]. X[z] is known to have a pole at z = (0.75+0.75j), and other poles may be present. Recall that an absolutely summable signal satisfies %^∞ −∞ |x[n]| < ∞.
  • (a) Can x[n] be left-sided? Explain.
  • (b) Can x[n] be right-sided? Explain.
  • (c) Can x[n] be two-sided? Explain.
  • (d) Can x[n] be of finite duration? Explain.
• 5.8-11 Consider a causal system that has transfer function

$$H[z] = \frac{z - 0.5}{z + 0.5}$$

When appropriate, assume initial conditions of zero.

• (a) Determine the output y1[n] of this system in response to *^x*1[n] = (3/4)nu[n].
• (b) Determine the output y2[n] of this system in response to *^x*2[n] = (3/4)n.
• (c) Determine the output y3[n] of this system in response to *^x*3[n] = (3/4)nu[−n−1].
• 5.8-12 Let ^x[n] =(−1)nu[n−n0]+αnu[−n]. Determine the constraints on the complex number α and the integer n⁰ so that the z-transform X[z] exists with region of convergence 1 < |z| < 2.
• 5.8-13 Using the definition, compute the bilateral z-transform, including the region of convergence (ROC), of the following complex-valued functions:
  • (a) *^x*1[n] = (−j)−nu[−n] +δ[−n]
  • (b) ^x2[n] = (j)ⁿ cos(n+1)u[n]
  • (c) x3[n] = jsinh(n)u[−n+1]
  • (d) ^x4[n] = %^{0 k}=−∞(2j)nδ[n−2k]
• 5.8-14 Use partial fraction expansions, z-transform tables, and a region of convergence (0.5 < |z| < 2) to determine the inverse z-transform of
  • (a) X1[z] = 1 ^{1+ 13 6 z}−^{1 + 1 6 z}−^{2 − 1 3} z−³ 1

(b)

$$X_2[z] = \frac{1}{z^{-3}(2 - z^{-1})(1 + 2z^{-1})}$$

5.8-15 Use partial fraction expansions, z-transform tables, and the fact that the systems are stable to determine the inverse z-transform of

(a)

$$H_1[z] = \frac{z^{-1}}{(z - \frac{1}{2})(1 + \frac{1}{2}z^{-1})}$$

(b)

$$H_2[z] = \frac{z+1}{z^3(z-2)(z+\frac{1}{2})}$$

5.8-16 By inserting N − 1 zeros between every sample of a unit step, we obtain a signal

$$h[n] = \sum_{k=0}^{\infty} \delta[n - Nk]$$

Determine H[z], the bilateral z-transform of h[n]. Identify the number and location(s) of the poles of H[z].

• 5.8-17 Using transform-domain techniques, determine the zero-state response yzsr[n] of LTID system ^y[n] − ^{1 4} y[n − 2] = x[n] to the noncausal input ^x[n] = ²nu[2−n].
• 5.8-18 Determine the zero-state response of a system having a transfer function

$$H[z] = \frac{z}{(z+0.2)(z-0.8)} \qquad |z| > 0.8$$

and an input x[n] given by

• (a) ^x[n] = enu[n]
• (b) ^x[n] = ²nu[−(n+1)]
• (c) ^x[n] = enu[n] +2nu[−(n+1)]
• (d) ^x[n] = ²nu[n] +u[−(n+1)]
• (e) ^x[n] = ^e−²nu[−(n+1)]
• 5.8-19 The discrete cross-correlation between real signal x[n] and real signal y[n] is

$$c_{xy}[n] = \sum_{k=-\infty}^{\infty} x[k]y[k-n]$$

Let signal x[n] have z-transform X[z] with ROC Rx, and let signal y[n] have z-transform Y[z] with ROC Ry. Determine CXY [z] (the bilateral z-transform of cxy[n]) in terms of the z-transforms of x[n] and y[n].

5.8-20 Transform properties can be very useful. The accumulation property states

$$\sum_{k=-\infty}^{n} x[k] \Longleftrightarrow \frac{z}{z-1} X[z]$$

This property is the DT version of the Laplace transform “integration in time” property. Given x[n] ⇐⇒ X[z], prove the accumulation property. [Hint: Polynomial long division can be helpful.] 5.10-1 Use MATLAB to generate pole-zero plots for the causal systems with the following transfer functions:

(a)

$$H_a[z] = \frac{z^4 - \sqrt{2}z^2 + 1}{z^4 + 0.4096}$$

(b) $H_b[z] = \frac{-3z^{-1} + \frac{3}{4}z^{-3}}{3 + \frac{3}{8}z^{-2} + 2z^{-4}}$

^{3+ 3 2 z}−2+2z−⁴ In each case, determine whether the system is

• stable.
• 5.10-2 For each of the following stable LTID systems, use MATLAB to generate magnitude and phase response plots over −π ≤ <π. (a) *^H*a[z] = cos(z) z−0.5
  • (b) Hb[z] = z³ sin(z−1)
• 5.10-3 Consider an LTID system described by the difference equation 4y[n + 2] − y[n] = x[n + 2] +x[n].
  • (a) Plot the pole-zero diagram for this system.
  • (b) Plot the system’s magnitude response ^|H[ej]| over ^{−π ≤ ≤ π}.
  • (c) What type of system is this: lowpass, highpass, bandpass, or bandstop?
  • (d) Is this system stable? Justify your answer.
  • (e) Is this system real? Justify your answer.
  • (f) If the system input is of the form x[n] = cos(n), what is the greatest possible amplitude of the output? Justify your answer.
  • (g) Draw an efficient, causal implementation of this system using only add, scale, and delay blocks.
• 5.10-4 Consider an LTID system with system function ^H(z) ^{= b0} *^z*2+¹ *^z*2−4/⁹ .
  • (a) Determine the constant b⁰ so that the system frequency response at = −π is −1.
  • (b) Plot the pole-zero diagram for this system.
  • (c) Plot the system’s magnitude response ^|H[ej]| over ^{−π ≤ ≤ π}.
  • (d) Plot the system’s phase response ^H[ej] over −π ≤ ≤ π.
  • (e) What type of system is this: lowpass, highpass, bandpass, or bandstop?
  • (f) Determine the response y[n] to the input ^x[n] = (−^{1 − j}) ⁺ (−j)^{n +} (^{1 − j}) cos(π*ⁿ* ^{+ 1} 3 ).
  • (g) Draw a TDFII block diagram representation of this system.

5.10-5 Consider the LTID system shown in Fig. P5.10-5, where parameters c¹ and c² are arbitrary constants.

Figure P5.10-5

• (a) Determine the system function H[z], expressed in standard rational form.
• (b) What is the order N of this system?
• (c) Determine the N poles and N zeros of this system. Use MATLAB to create the corresponding pole-zero plot.
• (d) What constraints, if any, exist on parameters c¹ and c² to ensure that the system is stable?
• (e) Determine c¹ and c² so that this system functions as an LPF with narrow passband. Use MATLAB to generate the corresponding magnitude response ^|H[ej]| over −π ≤ ≤ π.
• (f) Ms. Zeroine, the heroine of DT systems, believes that if x[n] = 0, then y[n] = 0 also. Is Ms. Zeroine correct? Fully justify your answer.
• (g) Dr. Strange suggests that by setting c¹ = −1 and c² = −2, the system will act as a highpass filter. Is Dr. Strange right? Fully justify your answer.
• 5.10-6 One interesting and useful application of discrete systems is the implementation of complex (rather than real) systems. A complex system is one in which a real-valued input can produce a complex-valued output. Complex systems that are described by constant coefficient difference equations require at least one complex-valued coefficient, and they are capable of operating on complex-valued inputs. Consider the complex discrete-time system

$$H[z] = \frac{z^2 - j}{z - 0.9e^{j3\pi/4}}$$

(a) Determine and plot the system zeros and poles.

• (b) Sketch the magnitude response ^|H[ejω]| of this system over −2π ≤ ω ≤ 2π. Comment on the system’s behavior.
• 5.10-7 Consider the complex system

$$H[z] = \frac{z^4 - 1}{2(z^2 + 0.81j)}$$

Refer to Prob. 5.10-6 for an introduction to complex systems.

• (a) Plot the pole-zero diagram for H[z].
• (b) Plot the system’s magnitude response ^|H[ej]| over ^{−π ≤ ≤ π}.
• (c) Explain why H[z] is a noncausal system. Do not give a general definition of causality; specifically identify what makes this system noncausal.
• (d) One way to make this system causal is to add two poles to H[z]. That is,

$$H_{\text{causal}}[z] = H[z] \frac{1}{(z-a)(z-b)}$$

Find poles ^a and ^b such that ^|Hcausal[ej]| = ^|H[ej]|.

• (e) Draw an efficient block implementation of Hcausal[z].
• 5.10-8 A discrete-time LTI system is shown in Fig. P5.10-8.
  • (a) Determine the difference equation that describes this system.
  • (b) Determine the magnitude response ^|H[ej]| for this system and simplify your answer. Plot the magnitude response over −π ≤ ≤ π. What type of standard filter (lowpass, highpass, bandpass, or bandstop) best describes this system?
  • (c) Determine the impulse response h[n] of this system.

Figure P5.10-8

• 5.10-9 Determine the impulse response h[n] for the system shown in Fig. P5.10-9. Is the system stable? Is the system causal?
• 5.10-10 An LTID filter has an impulse response function given by h[n] = δ[n − 1] + δ[n + 1]. Determine and carefully sketch the magnitude response ^|H[ej]| over the range ^{−π ≤ ≤ π}. For this range of frequencies, is this filter lowpass, highpass, bandpass, or bandstop?
• 5.10-11 A causal, stable discrete system has the rather strange transfer function H[z] = cos(z−1).
  • (a) Write MATLAB code that will compute and plot the magnitude response of this system over an appropriate range of digital frequencies . Comment on the system.
  • (b) Determine the impulse response h[n]. Plot h[n] over (0 ≤ n ≤ 10).
  • (c) Determine a difference equation description for an FIR filter that closely approximates the system H[z] = cos(z−1). To verify proper behavior, plot the FIR filter’s magnitude response and compare it with the magnitude response computed in part (a).
• 5.10-12 The MATLAB signal-processing toolbox function butter helps design digital Butterworth filters. Use MATLAB help to learn how butter works. For each of the following cases, design the filter, plot the filter’s poles and zeros in the complex z plane, and plot the decibel magnitude response 20log10 ^|H[ej]|.
  • (a) Design an eighth-order digital lowpass filter with ^c = π/3.
  • (b) Design an eighth-order digital highpass filter with ^c = π/3.
  • (c) Design an eighth-order digital bandpass filter with passband between 5π/24 and 11π/24.
  • (d) Design an eighth-order digital bandstop filter with stopband between 5π/24 and 11π/24.
• 5.10-13 The MATLAB signal-processing toolbox function cheby1 helps design digital Chebyshev

type I filters. A Chebyshev type I filter has passband ripple and smooth stopband. Setting the passband ripple to Rp = 3 dB, repeat Prob. 5.10-12 using the cheby1 command. With all other parameters held constant, what is the general effect of reducing Rp, the allowable passband ripple?

5.10-14 The MATLAB signal-processing toolbox function cheby2 helps design digital Chebyshev type II filters. A Chebyshev type II filter has smooth passband and ripple in the stopband. Setting the stopband ripple Rs = 20 dB down, repeat Prob. 5.10-12 using the cheby2 command. With all other parameters held constant, what is the general effect of increasing Rs, the minimum stopband attenuation?

5.10-15 The MATLAB signal-processing toolbox function ellip helps design digital elliptic filters. An elliptic filter has ripple in both the passband and the stopband. Setting the passband ripple to Rp = 3 dB and the stopband ripple Rs = 20 dB down, repeat Prob. 5.10-12 using the ellip command.