The bilinear transformation, introduced in the previous problem, may also be used to

Determine the constraint on r = \z\ for each of the following sums to converge: 10.2. (a) ~ (~)n+1 -n (b) ~(~)-n+lzn n=-1 n=l (c) ~{ l+(~l)"}z-n (d) ~ (4)1n1 cos(in)z-n

Consider the signal ( 1 )II x[n] = S u[n - 3]. Use eq. (10.3) to evaluate the z-transform of this signal, and specify the corresponding region of convergence.

Let Determine the constraints on the complex number a and the integer n0 , given that the ROC of X(z) is 10.4. Consider the signal 1 < \z\ < 2.

Consider the signal 1 < \z\ < 2. x[n] = { C1)n cos( in), 0, Determine the poles and ROC for X(z).

For each of the following algebraic expressions for the z-transform of a signal, determine the number of zeros in the finite z-plane and the number of zeros at infinity. 2(1- z-1) (c) (1- z- 1)(1 + z- 1)

Let x[n] be an absolutely summable signal with rational z-transform X(z). If X(z) is known to have a pole at z = 112, could x[n] be (a) a finite-duration signal? (b) a left -sided signal? (c) a right-sided signal? (d) a two-sided signal?

Suppose that the algebraic expression for the z-transform of x[n] is How many different regions of convergence could correspond to X(z)?

Let x[n] be a signal whose rational z-transform X(z) contains a pole at z = 112. Given that x 1 [n] ~ (U x[n] is absolutely summable and ( 1 )fl x2[n] = S x[n] is not absolutely summable, determine whether x[n] is left sided, right sided, or two sided.

Using partial-fraction expansion and the fact that z 1 anu[n] ~ 1 _1 , lzl > lal, - az find the inverse z-transform of

Find the inverse z-transform of 1 [ 1' 024 - z10 ] X(z) = 1,024 1 - ~z-1 , lzl > 0.

By considering the geometric interpretation of the magnitude of the Fourier transform from the pole-zero plot, determine, for each of the following z-transforms, whether the corresponding signal has an approximately lowpass, bandpass, or highpass characteristic: -1 (a) X(z) = z 8 _ 1 , lzl > ~ 1 + gZ 1 + __z- 1 (b) X(z) = 9 lzl > ~ 1 - ~z- 1 + ~z- ' 9 81 1 I 8 (c) X(z) = 1 64 _ 2 , zl > 9

Consider the rectangular signal x[n] = { ~ Let O::sn::s5 otherwise g[n] = x[n] - x[n - 1]. (a) Find the signal g[n] and directly evaluate its z-transform. (b) Noting that n x[n] = L g[k], k=-X use Table 10.1 to determine the z-transform of x[n].

Consider the triangular signal { n-1 g[n] = 13- ~. 0, (a) Determine the value of no such that 2::sn::s7 8 :::; n :::; 12 . otherwise g[n] = x[n] * x[n - no], where x[n] is the rectangular signal considered in Problem 10.13. (b) Use the convolution and shift properties in conjunction with X(z) found in Problem 10.13 to determine G(z). Verify that your answer satisfies the initialvalue theorem.

Let ( 1 )II y[n] = g u[n]. Determine two distinct signals such that each has a z-transform X(z) which satisfies both of the following conditions: 1. [X(z) +X(- z)]/2 = Y(z2). 2. X(z) has only one pole and only one zero in the z-plane.

Consider the following system functions for stable LTI systems. Without utilizing the inverse z-transform, determine in each case whether or not the corresponding system is causal. 1 4 -1 + I -2 (a) - 3z 2z z- 1(1- 4z- 1)(1- ~z- ) (b) (c) z-! 2 2 I _ 3 z + 2z 16 z + 1 z +:!- !z-2 - ~z- 3

Suppose we are given the following five facts about a particular LTI systemS with impulse response h[n] and z-transform H(z): 1. h[n] is real. 2. h[n] is right sided. 3. limH(z) = 1. z---->oo 4. H(z) has two zeros. 5. H (z) has one of its poles at a nonreallocation on the circle defined by lzl = 3/4. Answer the following two questions: (a) IsS causal? (b) IsS stable?

Consider a causal LTI system whose input x[n] and output y[n] are related through the block diagram representation shown in Figure Pl0.18. x[n] ~8-----~l----~~~ y[n] Figure P1 0.18 (a) Determine a difference equation relating y[n] and x[n]. (b) Is this system stable?

Determine the unilateral z-transform of each of the following signals, and specify the corresponding regions of convergence: (a) \(x_{1}[n]=\left(\frac{1}{4}\right)^{n} u[n+5]\) (b) \(x_{2}[n]=\delta[n+3]+\delta[n]+2^{n} u[-n]\) (c) \(x_{3}[n]=\left(\frac{1}{2}\right)^{|n|}\)

Determine the z-transform for each of the following sequences. Sketch the polezero plot and indicate the region of convergence. Indicate whether or not the Fourier transform of the sequence exists. (a) S[n + 5] (b) S[n- 5] (c) ( -l)nu[n] (d) C4Y+ 1 u[n + 3] (e) (-~)nu[-n- 2] (0 Ci)nu[3- n] (g) 2nu[ -n] + (i)nu[n- 1] (h) (~)n- u[n- 2]

Determine the z-transform for the following sequences. Express all sums in closed form. Sketch the pole-zero plot and indicate the region of convergence. Indicate whether the Fourier transform of the sequence exists. (a) (4)n{u[n + 4]- u[n- 5]} (b) n(4)1nl (c) lnl(4)1nl (d) 4n case; n + *]u[ -n- 1]

Following are several z-transforms. For each one, determine the inverse z-transform using both the method based on the partial-fraction expansion and the Taylor's series method based on the use of long division. 1- z- 1 1 X(z) = 1-.!.z-2' lzl > 2 4 1- z- 1 1 X(z) = 1- .!.z-2' lzl < 2 4 z-1-.!. 1 X(z) = 2 lzl > 2 1- .!.z- 1' 2 z-1-.!. 1 X(z) = 2 lzl < 2 1- .!.z- 1' 2 X(z) = z- 1 -.!. 1 (1 - 4z-~)2' lzl > 2. X(z) = z- 1 -.!. 1 (1- 4z-~)2' lzl < 2.

Using the method indicated, determine the sequence that goes with each of the following z-transforms: (a) Partial fractions: X(z) = 1 + ~z-I + z_ 2, and x[n] is absolutely summable. (b) Long division: 1 - !z- 1 X(z) = 2 and x[n] is right sided. 1 + !z- 1' 2 (c) Partial fractions: X(z) = 1 3 1 _ , and x[n] is absolutely summable

Consider a right-sided sequence x[n] with z-transform 1 X(z) = 1 ( 1 - 2 z-1 )(I - z-1) (P10.25-1) (a) Carry out a partial-fraction expansion of eq. (P10.25-1) expressed as a ratio of polynomials in z- 1, and from this expansion, determine x[n]. (b) Rewrite eq. (P10.25-1) as a ratio of polynomials in z, and carry out a partialfraction expansion of X(z) expressed in terms of polynomials in z. From this expansion, determine x[n], and demonstrate that the sequence obtained is identical to that obtained in part (a).

Consider a left-sided sequence x[n] with z-transform 1 X(z) = . ( 1 - ~ z-1 )(1 - z- I) (a) Write X(z) as a ratio of polynomials in z instead of z- 1 (b) Using a partial-fraction expression, express X(z) as a sum of terms, where each term represents a pole from your answer in part (a). (c) Determine x[n].

A right-sided sequence x[n] has z-transform X(z) = 3z- 10 + z- 7 - sz- 2 + 4z-l + 1 Determine x[n] for n < 0.

(a) Determine the z-transform of the sequence x[n] = B[n] - 0.95 B[n- 6]. (b) Sketch the pole-zero pattern for the sequence in part (a). (c) By considering the behavior of the pole and zero vectors as the unit circle is traversed, develop an approximate sketch of the magnitude of the Fourier transform of x[n].

By considering the geometric determination of the frequency response as discussed in Section 10.4, sketch, for each of the pole-zero plots in Figure P10.29, the magnitude of the associated Fourier transform.

We are given the following five facts about a discrete-time signal x[n] with ztransform X(z): 1. x[n] is real and right-sided. 2. X(z) has exactly two poles. 3. X(z) has two zeros at the origin. 4. X(z) has a pole at z = ~ej7T13 5. X(l) = ~ Determine X(z) and specify its region of convergence.

Consider an LTI system with impulse response and input h[n] = { ~~ n2:0 n

(a) Determine the system function for the causal LTI system with difference equation 1 1 y[n]- 2 y[n- 1] + 4 y[n- 2] = x[n]. (b) Using z-transforms, determine y[n] if x[n] = (4 )" u[n].

A causal LTI system is described by the difference equation y[n] = y[n- 1] + y[n- 2] + x[n- 1]. 805 (a) Find the system function H(z) = Y(z)/X(z) for this system. Plot the poles and zeros of H(z) and indicate the region of convergence. (b) Find the unit sample response of the system. (c) You should have found the system to be unstable. Find a stable (noncausal) unit sample response that satisfies the difference equation.

Consider an LTI system with input x[n] and output y[n] for which 5 y[n - 1] - 2 y[n] + y[n + 1] = x[n]. The system may or may not be stable or causal. By considering the pole-zero pattern associated with the preceding difference equation, determine three possible choices for the unit sample response of the system. Show that each choice satisfies the difference equation.

Consider the linear, discrete-time, shift-invariant system with input x[ n] and output y[n] for which \(y[n-1]-\frac{10}{3} y[n]+y[n+1]=x[n]\) The system is stable. Determine the unit sample response.

The input x[n] and output y[n] of a causal LTI system are related through the block-diagram representation shown in Figure Pl0.37. (a) Determine a difference equation relating y[n] and x[n]. (b) Is this system stable?

Consider a causal LTI systemS with input x[n] and a system function specified as H(z) = H1 (z)H2(z), where and A block diagram corresponding to H(z) may be obtained as a cascade connection of a block diagram for H 1 (z) followed by a block diagram for H2(z). The result is shown in Figure PI 0.38, in which we have also labeled the intermediate signals e1 [n], e2[n], !1 [n], and f2[n]. (a) How is e1 [n] related to / 1 [n]? (b) How is e2[n] related to f2[n]? y[n] Figure P1 0.38 (c) Using your answers to the previous two parts as a guide, construct a directform block diagram for S that contains only two delay elements. (d) Draw a cascade-form block diagram representation for S based on the observation that H(z) = 4 ( 1 + lz- 1 )(1- 2z-I) 1 + ~ z- 1 1 - ~ z- 1 (e) Draw a parallel-form block diagram representation for S based on the observation that 5/3 H(z) = 4 + ----=--- 1 + lz-I 2 14/3 1- lz-I.

Consider the following three system functions corresponding to causal LTI systems: H1 (z) = ---------,-----=------ (1 - z- 1 + ~z- )(1 - ~z- 1 + ~z- )' 1 H2(z) = -----,------,------- (1- z-I + ~z-2)(1 - ~z-I + z-2)' 1 H3(z) = ---------,------- (1- z-I + ~z-2)(1- z-I + ~z-2) (a) For each system function, draw a direct-form block diagram. (b) For each system function, draw a block diagram that corresponds to the cascade connection of two second-order block diagrams. Each second-order block diagram should be in direct form. (c) For each system function, determine whether there exists a block diagram representation which is the cascade of four first-order block diagrams with the constraint that all the coefficient multipliers must be real.

Consider the following two signals: ( 1 )n+ I XJ [n] = 2 u[n + 1], ( 1 )n x2[n] = 4 u[n]. Let X 1 (z) and X1 (z) respectively be the unilateral and bilateral z-transforms of x 1 [n], and let X2(z) and X2(z) respectively be the unilateral and bilateral ztransforms of x2 [ n]. (a) Take the inverse bilateral z-transform of X1 (z)X2(z) to determine g[n] = XJ [n] * x2[n]. (b) Take the inverse unilateral z-transform ofX1(z)X2(z) to obtain a signal q[n] for n 2: 0. Observe that q[n] and g[n] are not identical for n 2: 0.

For each of the following difference equations and associated input and initial conditions, determine the zero-input and zero-state responses by using the unilateral z-transform: (a) y[n] + 3y[n- 1] = x[n], ( 1 )ll x[n] = 2 u[n], y[-1] = 1. 1 1 (b) y[n] - 2 y[n- 1] = x[n] - 2 x[n- 1], x[n] = u[n], y[-1]=0. 1 1 (c) y[n]- 2 y[n- 1] = x[n]- 2x[n- 1], x[n] = u[n], y[-1] = 1.

Consider an even sequence x[n] (i.e., x[n] = x[ -n]) with rational z-transform X(z). (a) From the definition of the z-transform, show that X(z) = x(H (b) From your results in part (a), show that if a pole (zero) of X(z) occurs at z = zo, then a pole (zero) must also occur at z = 1/z0 . (c) Verify the result in part (b) for each of the following sequences: (1) o[n + 1] + o[n - 1] (2) o[n + 1] - ~o[n] + o[n- 1]

Let x[n] be a discrete-time signal with z-transform X(z). For each of the following signals, determine the z-transform in terms of X(z): (a) Llx[n], where Ll is the first-difference operator defined by Llx[n] = x[n] - x[n- 1] (b) XI [n] = { x[n/2], 0, (c) x 1 [n] = x[2n]

Determine which of the following z-transforms could be the transfer function of a discrete-time linear system that is not necessarily stable, but for which the unit sample response is zero for n < 0. State your reasons clearly. (1- z-1)2 (b) (z- 1)2 (a) 1 - .! z- 1 z - .! 2 2 (z- ~)5 (z- ~)6 (c) (d) (z - ~ )6 (z - ~ )5

A sequence x[n] is the output of an LTI system whose input is s[n]. The system is described by the difference equation x[n] = s[n] - e8 a s[n - 8], where 0 < a < 1. (a) Find the system function X(z) H1 (z) = S(z), and plot its poles and zeros in the z-plane. Indicate the region of convergence. (b) We wish to recover s[n] from x[n] with an LTI system. Find the system function H ( ) = Y(z) 2 z X(z) such that y[n] = s[n]. Find all possible regions of convergence for H2(z), and for each, tell whether or not the system is causal or stable. (c) Find all possible choices for the unit impulse response h2 [n] such that y[n] = h2[n] * x[n] = s[n].

The following is known about a discrete-time LTI system with input x[n] and output y[n]: 1. If x[n] = ( -2)n for all n, then y[n] = 0 for all n. 2. If x[n] = (112)nu[n] for all n, then y[n] for all n is of the form y[n] = ll[n] +a(~)" u[n], where a is a constant. (a) Determine the value of the constant a. (b) Determine the response y[n] if the input x[n] is x[n] = 1, for all n.

Suppose a second-order causal LTI system has been designed with a real impulse response hi [n] and a rational system function HI (z). The pole-zero plot for HI (z) is shown in Figure P10.48(a). Now consider another causal second-order system with impulse response h2 [n] and rational system function H2 (z). The pole-zero plot for H2(z) is shown in Figure P10.48(b). Determine a sequence g[n] such that the following three conditions hold: (1) h2[n] = g[n]h 1 [n] (2) g[n] = 0 for n < 0 (3) _Lig[kJI = 3 k=O

In Property 4 of Section 10.2, it was stated that if x[n] is a right-sided sequence and if the circle lzl = r0 is in the ROC, then all finite values of z for which 1z1 > r0 will also be in the ROC. In this discussion an intuitive explanation was given. A more formal argument parallels closely that used for Property 4 of Section 9 .2, relating to the Laplace transform. Specifically, consider a right-sided sequence x[n] = 0, n < NJ, and for which L, ix[n]jr0n L, jx[n]jr0n < oo. n=-oo n=N1 Then if ro :::; r1, (P10.49-1) where A is a positive constant. (a) Show that eq. (Pl0.49-l) is true, and determine the constant A in terms of r0 , r 1, andN1 (b) From your result in part (a), show that Property 4 of Section 10.2 follows. (c) Develop an argument similar to the foregoing one to demonstrate the validity of Property 5 of Section 1 0.2.

Consider a real-valued sequence x[n] with rational z-transform X(z). (a) From the definition of the z-transform, show that X(z) = X*(z*). (b) From your result in part (a), show that if a pole (zero) of X(z) occurs at z = z0 , then a pole (zero) must also occur at z = z~. (c) Verify the result in part (b) for each of the following sequences: (1) x[n] = (4)nu[n] (2) x[n] = o[n] - 4o[n- 1] + io[n- 2] (d) By combining your results in part (b) with the result of Problem 10.43(b), show that for a real, even sequence, if there is a pole (zero) of H (z) at z = peF1, then there is also a pole (zero) of H(z) at z = (11 p)ei8 and at z = (11 p)e-i 8 .

Consider a sequence xdn] with z-transform X1 (z) and a sequence x 2 [n] with ztransform X2(z), where Show that X2(z) = X1 (11z), and from this, show that if X1 (z) has a pole (or zero) at z = zo, then X2(z) has a pole (or zero) at z = 1/zo.

(a) Carry out the proof for each of the following properties in Table 10.1: (1) Property set forth in Section 10.5.2 (2) Property set forth in Section 10.5.3 (3) Property set forth in Section 10.5.4 (b) WithX(z) denoting the z-trarisform of x[n] and Rx the ROC of X(z), determine, in terms of X(z) and Rx. the z-transform and associated ROC for each of the following sequences: (1) x*[n] (2), z0x[n], where zo is a complex number

In Section 10.5.9, we stated and proved the initial-value theorem for causal sequences. (a) State and prove the corresponding theorem if x[n] is anticausal (i.e., if x[n] = 0, n > 0). (b) Show that if x[n] = 0, n < 0, then x[1] = lim z(X(z) - x[O]).

Let x[n] denote a causal sequence (i.e., if x[n] = 0, n < 0) for which x[O] is nonzero and finite. (a) Using the initial-value theorem, show that there are no poles or zeros of X(z) at z = oo. (b) Show that, as a consequence of your result in part (a), the number of poles of X(z) in the finite z-plane equals the number of zeros of X(z) in the finite z-plane. (The finite z-plane excludes z = oo.)

In Section 10.5.7, we stated the convolution property for the z-transform. To show that this property holds, we begin with the convolution sum expressed as 00 X3[n] = XI [n] * X2[n] = ~ XI [k]x2[n- k]. (P10.56-l) k= -00 (a) By taking the z-transform of eq. (P10.56-1) and using eq. (10.3), show that X3(Z) = ~ XI [k]X2(Z), k= -00 where X2(z) is the transform of x2 [n - k]. (b) Using your result in part (a) and property 10.5.2 in Table 10.1, show that 00 X3(z) = X2(z) ~ xi [k]z-k. (c) From part (b), show that as stated in eq. (10.81).

Let k= -00 XI(z) = xi[O] + xi[1]z-I + + xi[NJ]z-N', X2(z) = x2[0] + x2[l]z-I + + x2[N2]z-N2 Define and let M f(z) = L, y[k]z-k. k=O (a) Express Min terms of N1 and N2. (b) Use polynomial multiplication to determine y[O], y[l], and y[2]. (c) Use polynomial multiplication to show that, for 0 :::; k :::; M, y[k] = L, XI [m]x2[k- m].

A minimum-phase system is a system that is causal and stable and for which the inverse system is also causal and stable. Determine the necessary constraints on the location in the z-plane of the poles and zeros of the system function of a minimumphase system.

Consider the digital filter structure shown in Figure P10.59. k 3 k 4 Figure P1 0.59 (a) Find H(z) for this causal filter. Plot the pole-zero pattern and indicate theregion of convergence. (b) For what values of the k is the system stable? (c) Determine y[n] if k = 1 and x[n] = (2/3)n for all n.

If~(z) denotes the unilateral z-transform of x[n], determine, in terms of~(z), the unilateral z-transform of: (a) x[n + 3] (b) x[n - 3] (c) L ~= _ 00 x[k]

The autocorrelation sequence of a sequence x[n] is defined as \(\phi_{x x}[n]=\sum_{k=-\infty}^{\infty} x[k] x[n+k]\) Determine the z-transform of \(\phi_{x x}[n]\) in terms of the z-transform of x[n]

By using the power-series expansion oo wi log(l - w) = - L --:-, lwl < 1, i = 1 l determine the inverse of each of the following two z-transforms: (a) X(z) = log(l - 2z), lzl < 1 (b) X(z) = log(l -1z- 1 ), lzl > 1

By first differentiating X(z) and using the appropriate properties of the z-transform, determine the sequence for which the z-transform is each of the following: (a) X(z) = log(l - 2z), lzl < 1 (b) X(z) = log(l-1z- 1), lzl > 1 Compare your results for (a) and (b) with the results obtained in Problem 10.63, in which the power-series expansion was used.

The bilinear transformation is a mapping for obtaining a rational z-transform Hd(Z) from a rational Laplace transform Hc(s). This mapping has two important properties: 1. If Hc(s) is the Laplace transform of a causal and stable LTI system, then Hd(Z) is the z-transform of a causal and stable LTI system. 2. Certain important characteristics of \(\left|H_{c}(j \omega)\right|\) are preserved in \(\left|H_{d}\left(e^{j \omega}\right)\right|\). In this problem, we illustrate the second of these properties for the case of all-pass filters. (a) Let \(H_{c}(s)=\frac{a-s}{s+a} \text {, }\) where a is real and positive. Show that \(\left|H_{c}(j \omega)\right|=1\) (b) Let us now apply the bilinear transformation to Hc(s) in order to obtain Hd(Z). That is, \(H_{d}(z)=\left.H_{c}(s)\right|_{s=\frac{1-z^{-1}}{1+z^{-1}}}\) Show that Hd(Z) has one pole (which is inside the unit circle) and one zero (which is outside the unit circle). (c) For the system function Hd(Z) derived in part (b), show that \(\left|H_{d}\left(e^{j \omega}\right)\right|=1\).

The bilinear transformation, introduced in the previous problem, may also be used to obtain a discrete-time filter, the magnitude of whose frequency response is similar to the magnitude of the frequency response of a given continuous-time lowpass filter. In this problem, we illustrate the similarity through the example of a continuous-time second-order Butterworth filter with system function Hc(s). (a) Let Hd(Z) = Hc(s)Js= 1-z-1 l+z-1 Show that (b) Given that 1 Hc(s) = 14 14 (s + e/rr )(s + e- j1r ) and that the corresponding filter is causal, verify that Hc(O) = 1, that IHc(Jw )J decreases monotonically with increasing positive values of w, that IHc(j)J 2 = 112 (i.e., that We = 1 is the half-power frequency), and that Hc(oo) = 0. (c) Show that if the bilinear transformation is applied to Hc(s) of part (b) in order to obtain Hd(Z), then the following may be asserted about Hd(Z) and Hd(ejw): 1. Hd(Z) has only two poles, both of which are inside the unit circle. 2. Hd(ej0 ) = 1. 3. IHd(ejw)l decreases monotonically as w goes from 0 to TT. 4. The half-power frequency of Hd(ejw) is TT/2.