Although the history of the Z-transform is originally connected with probability theory, for discrete- time signals and systems it can be connected with the Laplace transform. The periodicity in the frequency domain and the possibility of an infinite number of poles and zeros makes this connec- tion not very useful. Defining a new complex variable in polar form provides the definition of the Z-transform and thez-plane. As with the Laplace transform, poles of the Z-transform characterize discrete-time signals by means of frequency and attenuation. One- and two-sided Z-transforms are possible, although the one-sided version can be used to obtain the two-sided one. The region of con- vergence makes the Z-transform have a unique relationship with the signal, and it will be useful in obtaining the discrete Fourier representations in Chapter 10.
Dynamic systems represented by difference equations use the Z-transform for representation by means of the transfer function. The one-sided Z-transform is useful in the solution of difference equations with nonzero initial conditions. As in the continuous-time case, filters can be represented by difference equations. However, discrete filters represented by polynomials are also possible. These nonrecursive filters give significance to the convolution sum, and will motivate us to develop methods that efficiently compute it.
You will see that the reason to present the Z-transform before the Fourier representation of discrete- time signals and systems is to use their connection, thereby simplifying calculations.
PROBLEMS
9.1. Mapping ofs-plane into thez-plane
The poles of the Laplace transformX(s)of an analog signalx(t)are p1,2= −1±j1 p3=0 p4,5= ±j1
There are no zeros. If we use the transformationz=esTswithTs=1:
(a) Determine where the given poles are mapped into thez-plane.
(b) How would you determine if these poles are mapped inside, on, or outside the unit circle in thez- plane? Explain.
(c) Carefully plot the poles and the zeros of the analog and the discrete-time signals in the Laplace and thez-planes.
9.2. Mapping ofz-plane into thes-plane
Consider the inverse relation given byz=esTs—that is, how to map thez-plane into thes-plane.
(a) Find an expression forsin terms ofzfrom the relationz=esTs.
(b) Consider the mapping of the unit circle (i.e.,z=1ejω,−π≤ω < π). Obtain the segment in thes-plane resulting from the mapping.
(c) Consider the mapping of the inside and the outside of the unit circle. Determine the regions in the s-plane resulting from the mappings.
Problems 565
(d) From the above results, indicate the region in thes-plane to which the wholez-plane is mapped into.
Sinceω=ω+2π, is this mapping unique? Explain.
9.3. Z-transform and ROCs Consider the noncausal sequence
s[n]=s1[n]+s2[n]
wheres1[n]=u[n]is causal ands2[n]= −u[−n]is anti-causal. This signal is thesignum, or sign function, that extracts the sign of a real-valued signal—that is,
s[n]=sgn(x[n])=
−1 x[n]<0 0 x[n]=0 1 x[n]>0
(a) Find the Z-transforms ofs1[n]ands2[n], indicating the corresponding ROC.
(b) Determine the Z-transformS(z). 9.4. Z-transform and ROC
Given the anti-causal signal
x[n]= −αnu[−n]
(a) Determine the Z-transformX(z), and carefully plot the ROC whenα=0.5andα=2. For which of the two values ofαdoesX(ejω)exist?
(b) Find the signal that corresponds to the derivativedX(z)/dz. Express it in terms ofα.
9.5. Significance of ROC
Consider a causal signalx1[n]=u[n]and an anti-causal signalx2[n]= −u[−n−1].
(a) Find the Z-transformsX1(z)andX2(z)and carefully plot their ROCs. If the ROCs are not included with the Z-transforms, would you be able to tell which is the correct inverse? Explain.
(b) Determine if it is possible to find the Z-transform ofx1[n]+x2[n].
9.6. Fibonacci sequence generation—MATLAB
Consider the Fibonacci sequence generated by the difference equation f[n]=f[n−1]+f[n−2] n≥0 with initial conditionsf[−1]=1,f[−2]= −1.
(a) Find the Z-transform off[n], orF(z).
(b) Find the polesφ1andφ2and the zeros ofF(z)and plot them. How are the poles connected? How are they related to the “golden ratio”?
(c) The Fibonacci difference equation has zero input, but its response is a sequence of ever-increasing integers. Obtain a partial fraction expansion ofF(z)and findf[n]in terms of the polesφ1andφ2, and show that the result is always integer. Use MATLAB to implement the inverse in term of the poles.
9.7. Laplace and Z-transforms of sampled signals
An analog pulsex(t)=u(t)−u(t−1)is sampled using a sampling periodTs=0.1.
(a) Obtain the discrete-time signalx(nTs)=x(t)|t=nTsand plot it as a function ofnTs. (b) If the sampled signal is represented as an analog signal as
xs(t)= N−1
X n=0
x(nTs)δ(t−nTs) determine the value ofNin the above equation.
(c) Compute the Laplace transform of the sampled signal (i.e.,Xs(s)=L[xs(t]).
(d) Determine the Z-transform ofx(nTs), orX(z). (e) Indicate how to transformXs(s)intoX(z) 9.8. Computation of Z-transform—MATLAB
Consider a discrete-time pulsex[n]=u[n]−u[n−10].
(a) Plotx[n]as a function ofnand use the definition of the Z-transform to findX(z).
(b) Use the Z-transform ofu[n]and properties of the Z-transform to findX(z). Verify that the expressions obtained above forX(z)are identical.
(c) Find the poles and the zeros ofX(z)and plot them in thez-plane. Use MATLAB to plot the poles and zeros.
9.9. Computation of Z-transform
A causal exponential x(t)=2e−2tu(t) is sampled using a sampling period Ts=1. The corresponding discrete-time signal isx[n]=2e−2nu[n].
(a) Express the discrete-time signal asx[n]=2αnu[n]and give the value ofα. (b) Find the Z-transformX(z)ofx[n]and plot its poles and zeros in thez-plane.
9.10. Computation of Z-transform
Consider the signalx[n]=0.5(1+[−1]n)u[n].
(a) Plotx[n]and use the definition of the Z-transform to obtain its Z-transform,X(z).
(b) Use the linearity property and the Z-transforms ofu[n]and[−1]nu[n]to find the Z-transformX(z)= Z[x[n]].
(c) Determine and plot the poles and the zeros ofX(z). 9.11. Solution of difference equations with Z-transform
Consider a system represented by the first-order difference equation y[n]=x[n]−0.5y[n−1]
wherey[n]is the output andx[n]is the input.
(a) Find the Z-transformY(z)in terms ofX(z)and the initial conditiony[−1].
(b) Find an inputx[n]6=0and an initial conditiony[−1]6=0so that the output isy[n]=0forn≥0. Verify you get this result by solving the difference equation recursively.
(c) For zero initial conditions, find the inputx[n]so thaty[n]=δ[n]+0.5δ[n−1].
9.12. Transfer function, stability, and impulse response—MATLAB
Consider a second-order discrete-time system represented by the difference equation y[n]−2rcos(ω0)y[n−1]+r2y[n−2]=x[n] n≥0 wherer>0and0≤ω0≤2π,y[n]is the output, andx[n]is the input.
(a) Find the transfer functionH(z)of this system.
(b) Find the value ofω0 and determine the values ofr that would make the system stable. Use the MATLAB functionzplaneto plot the poles and the zeros forr=0.5andω0=π/2radians.
(c) Letω0=π/2. Find the corresponding impulse responseh[n]of the system. What other value ofω0 would get the same impulse response?
9.13. Generation of discrete-time sinusoid—MATLAB
Given that the Z-transform of a discrete-time cosineAcos(ω0n)u[n]is A(1−cos(ω0)z−1) 1−2 cos(ω0)z−1+z−2
Problems 567
(a) Use the given Z-transform to find a difference equation for which the outputy[n]is a discrete-time cosineAcos(ω0n)and the input isx[n]=δ[n]. What should you use as initial conditions?
(b) Verify your algorithm by generating a signaly[n]=2 cos(πn/2)u[n]by implementing your algorithm in MATLAB. Plot the input and the output signalsx[n]andy[n].
(c) Indicate how to change your previous algorithm to generate a sine functiony[n]=2 sin(πn/2)u[n].
Use MATLAB to findy[n], and to plot it.
9.14. Inverse Z-transform and poles and zeros
When finding the inverse Z-transform of functions withz−1terms in the numerator, the fact thatz−1can be thought of as a delay operator can be used to simplify the computation. Consider
X(z)= 1−z−10 1−z−1
(a) Use the Z-transform ofu[n]and the properties of the Z-transform to findx[n].
(b) If we considerX(z)a polynomial in negative powers ofz, what would be its degree and the values of its coefficients?
(c) Find the poles and the zeros ofX(z)and plot them on thez-plane. Is there a pole or zero atz=1?
Explain.
9.15. Initial conditions and steady state
Consider a second-order system represented by the difference equation y[n]=0.25y[n−2]+x[n]
wherex[n]is the input andy[n]is the output.
(a) For the zero-input case (i.e., whenx[n]=0), find the initial conditionsy[−1]andy[−2]so thaty[n]= 0.5nu[n].
(b) Suppose the input isx[n]=u[n]. Without solving the difference equation can you find the correspond- ing steady stateyss[n]? Explain how and give the steady-state output. Verify by inverse Z-transform that the steady-state responseyss[n]is the one obtained.
9.16. Initial conditions and impulse response A second-order system has the difference equation
y[n]=0.25y[n−2]+x[n]
wherex[n]is the input andy[n]is the output.
(a) Find the inputx[n]so that for zero initial conditions, the output is given asy[n]=0.5nu[n].
(b) Ifx[n]=δ[n]+0.5δ[n−1]is the input to the above difference equation, find the impulse response of the system.
9.17. Convolution sum and product of polynomials
The convolution sum is a fast way to find the coefficients of the polynomial resulting from the multiplication of two polynomials.
(a) Supposex[n]=u[n]−u[n−3]. Find its Z-transformX(z), a second-order polynomial inz−1. (b) MultiplyX(z)by itself to get a new polynomialY(z)=X(z)X(z)=X2(z). FindY(z).
(c) Graphically show the convolution ofx[n]with itself and verify that the result coincides with the coefficients ofY(z).
9.18. Inverse Z-transform Find the inverse Z-transform of
X(z)= 8−4z−1 z−2+6z−1+8 and determinex[n]asn→ ∞. Assumex[n]is causal.
9.19. Z-transform properties and inverse transform
Sometimes the partial fraction expansion is not needed in finding the inverse Z-transform—instead the properties of the transform can be used. Consider the function
F(z)= z+1 z2(z−1)
(a) Determine whetherF(z)is a proper rational function as a function ofzand ofz−1. (b) Verify thatF(z)can be written as
F(z)= z−2
1−z−1 + z−3 1−z−1 Find the inverse Z-transformf[n]using the above expression.
9.20. Inverse Z-transform—MATLAB
We are interested in the unit-step solution of a system represented by the difference equation y[n]=y[n−1]−0.5y[n−2]+x[n]+x[n−1]
(a) Find an expression forY(z). (b) Do a partial expansion ofY(z).
(c) Find the inverse Z-transformy[n]and verify your results using MATLAB.
9.21. Pad ´e approximation
Suppose we are given a finite-length sequenceh[n](it could be part of an infinite-length impulse response from a discrete system that has been windowed) and would like to obtain a rational approximation for it.
This means that ifH(z)=Z[h[n]], a rational approximation of it would beH(z)=B(z)/A(z), from which we get
H(z)A(z)=B(z) Letting
B(z)= M−1
X k=0
bkz−k
A(z)=1+ N−1
X k=1
akz−k
for some choice ofM andN, equations from H(z)A(z)=B(z) should allow us to find the M+N−1 coefficients{akbk}.
Problems 569
(a) Find a matrix equation that would allow us to find the coefficients ofB(z)andA(z).
(b) Leth[n]=0.5n(u[n]−u[n−101])be the sequence we wish to obtain a rational approximation and letB(z)=b0whileA(z)=a0+a1z−1. Find the equations to solve for the coefficients{b0,a0,a1}. 9.22. Prony’s rational approximation—MATLAB
The Pad ´e approximants provide an exact matching ofM+N−1 values ofh[n]whereM andN are, respectively, the orders of the numerator and the denominator of the rational approximation. But there is no method for choosing the numerator and the denominator orders,MandN. Also, there is no guarantee on how well the rest of the signal is matched. Prony’s rational approximation considers how well the rest of the signal is approximated when finding the approximation. Leth[n]=0.9nu[n]be the exact impulse response for which we wish to find a rational approximation. Take the first100values of this signal as the impulse response.1
(a) Assume the order of the numerator and the denominator are equal, M=N=1. Use the MATLAB functionpronyto obtain the rational approximation, and then usefilterto verify that the impulse response of the rational approximation is close to the given100values. Plot the error between h[n]and the impulse response of the rational approximation for the first200samples.
(b) Plot the poles and the zeros of the rational approximation and compare them to the poles and the zeros ofH(z)=Z(h[n].
(c) Suppose thath[n]=(h1∗h2)[n]—that is, the convolution ofh1[n]=0.9nu[n]andh2[n]=0.8nu[n].
Use againpronyto find the rational approximation when the first 100 values ofh[n]are available. Use convfrom MATLAB to computeh[n]. Compare the impulse response of the rational approximation to h[n]. Plot the poles and the zeros ofH(z)=Z(h[n])and of the rational approximation.
(d) Consider theh[n]given above, and perform the Prony approximation using ordersM=N=3. Explain your results. Plot the poles and the zeros.
9.23. MATLAB partial fraction expansion
Consider the partial fraction expansion that MATLAB uses.
(a) Find the inverse Z-transform ofa/(1−az−1)2.
(b) Suppose that the partial fraction expansion given by MATLAB is X(z)= −1
1−0.5z−1 + 1 (1−0.5z−1)2 Determine the inversex[n].
9.24. MATLAB partial fraction expansion Consider finding the inverse Z-transform of
X(z)= 2z−1
(1−z−1)(1−2z−1)2 |z|>2 MATLAB does the partial fraction expansion as
X(z)= A
1−z−1 + B
1−2z−1 + C (1−2z−1)2 while we do it in the following form:
X(z)= D
1−z−1 + E
1−2z−1 + Fz−1 (1−2z−1)2 Show that the two partial fraction expansions give the same result.
1Gaspar de Prony (1765–1839) was a French mathematician and engineer, while Henri Pad´e (1863–1953) was a French mathemati- cian interested in rational approximations.
9.25. Prony method and Z-Transform—MATLAB
Consider finding the Z-transform of a noncausal signalh[n]=0.5nu[n+1]using the Prony approxima- tion.
(a) Use thepronyfunction to find a rational approximation forh[n](i.e., the Z-transformH(z)=B(z)/A(z)).
Use a first order for the numerator and the denominator.
(b) Separate the signal into its causal and anti-causal components, and usepronyto find the rational approximation of the causal and then add the anti-causal component to correct the above result.