EC2202: Analog and Digital Signal Processing

Tutorial 2

(January 20, 2011)

1. Consider the harmonically related periodic signals xk[n] = cos(2πkn/N), where 0 ≤k ≤ N − 1. Since xk[n] is real-valued, the highest value of k can be limited to theinteger that is less than or equal to N/2. What is the period of xk[n] for each k whenN = 11 and N = 12? Is there any difference in behaviour when compared with itscontinuous-time counterpart?

2. Is zn u[n] an eigensignal for an LTI system? Is cos(ω0n) an eigensignal? Is there acondition under which cos(ω0n) can be an eigensignal? Explain your answers.

3. Consider the non-causal rectangular window x[n] = 1 for −M ≤ n ≤ M. Derive

the expression for its DTFT X(ejω). Now make the window causal, i.e., x[n] = 1 for0 ≤ n ≤ 2M, and compute its DTFT. Can you express it in the form e−jKωA(ω), whereA(ω) is a purely real-valued function? What is the value of K ?

4. From the knowledge of the DTFTs of u[n] and cos(ω0n) findDTFT of x[n] = cos(ω0n) u[n].

5. Consider the pair x[n]DTFT←→ X(ejω). Find x[n] corresponding to (a) X(ejω) = j(4 +

2 cosω + 3 cos 2ω) · sin(ω/2) · ejω/2, (b) X(ejω) = cosN(ω) where N > 0, and (c)X(ejω) = cosN(ω/2) where N > 0 is an even integer.

6. The 2π-periodic function X(ejω) is defined as follows, where A > 0 :

X(ejω) =

A+ Aω

π−π ≤ ω < 0

−A+ Aω

π0 ≤ ω < π

(a) Derive the inverse DTFT of X(ejω).

(b) Using the above result derive the DTFT of x[n] = 1/n for n ≥ 1 and zero other-wise.

(c) Apply Parseval’s theorem and find the value of∞

∑n=1

1

n2.

7. Consider the real, positive function G(ejω) =1

A− B cosωwhere 0 < B < A. We wish

to find an absolutely summable sequence x[n] =1√αcnu[n] such that G(ejω) = |X(ejω)|2.

(a) By setting up a pair of equations, solve for α and c in terms of A and B.

(b) Using Parseval’s relation, obtain an expression for1

2π

∫ π

−πG(ejω) dω in terms of

A and B.

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8. The input to a cascade of two LTI systems is the signal x[n] = cos(0.6πn) + 3δ[n −5] + 2. The first system’s impulse response has the DTFT given by

H1(ejω) =

1 |ω| ≤ 0.5π0 0.5π < |ω| ≤ π

The second system is governed by the following input-output relationship: y[n] =w[n]−w[n− 1]. Find the output y[n].

9. Let h[n] = hR[n] + jhI [n] have DTFT H(ejω) = HR(ejω) + jHI(ejω), where the sub-

scripts R and I denote the real and imaginary parts. Let HER and HOR denote the evenand odd parts of HR; similarly, HEI and HOI are the even and odd parts of HI. If thetransform of hR[n] is Ha + jHb and that of hI [n] is Hc + jHd, express Ha, Hb, Hc, andHd in terms of HER, HOR, HEI, and HOI.

10. Let H(z) =−a∗ + z−1

1− az−1with RoC |z| > |a| and 0 < |a| < 1. Plot |H(ejω)| and comment

on the result.

11. Make a comparative table of continuous-time and discrete-time Fourier transformsof various commonly encountered CT signals and their discrete-time counterparts:δ(t) 1 δ[n], u(t) 1 u[n], cos(Ω0t) 1 cos(ω0n), sgn(t) 1 sgn[n] (sgn[0] = 1), e−atu(t) 1

anu[n], u(t + T)− u(t− T) 1 u[n + M]− u[n−M].

12. For the Xi(ω) given below, find the corresponding xi[n]. Note that the notation Xi(ω)has been used in this problem, rather than the usual Xi(e

jω)

π

π

−π

−π

X1(ω) X2(ω)

ω

ω

1

1

−1

13. Let X1(ejω) = 1 for |ω| ≤ π/4 and X1(e

jω) = X2(ejω). Plot the result of circularly

convolving X1(ejω) and X2(e

jω). Now if the cutoff frequency of the ideal LPF X1(ejω)

is π/2 and that of X2(ejω) is 3π/4, how will the above convolution result change?

Let X1(Ω) and X2(Ω) be the aperiodic versions of X1(ejω) and X2(e

jω). For both cases,linearly convolve X1(Ω) and X2(Ω), and take its result to form a periodic signal withperiod 2π. This can be done by shifting the linear convolution result by 2πk andadding over all k. How does this compare with the result of circular convolution? Canyou now relate linear and circular convolution? Commonly used notation for circularconvolution is©∗ .

14. Let

y[n] =

(

sin π4 n

πn

)2

∗(

sinωcn

πn

)

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where ∗ denotes convolution and |ωc| ≤ π. Determine a stricter constraint on ωc

which ensures that

y[n] =

(

sin π4 n

πn

)2

15. Let the inverse Fourier transform of Y(ejω) be

y[n] =

(

sinωcn

πn

)2

where 0 < ωc < π. Determine the value of ωc which ensures that Y(ejπ) = 1/2.

16. The DTFT of a particular signal is

X(ejω) =3

∑k=0

(1/2)k

1− 14 e−j(ω−kπ/2)

It can be shown that x[n] = g[n]q[n], where g[n] is of the form αnu[n] and q[n] is aperiodic signal with period N. (a) Determine the value of α. (b) Determine the valueof N. (c) Is x[n] real-valued?

17. Determine and sketch the DTFT Xi(ω) of xi[n]:

(a) x1[n] = 1, 1,↓1, 1, 1

(b) x2[n] = 1, 0, 1, 0,↓1, 0, 1, 0, 1

(c) x3[n] = 1, 0, 0, 1, 0, 0,↓1, 0, 0, 1, 0, 0, 1

Is there any relation between X1(ω), X2(ω), and X3(ω)?

18. Consider and LTI system with frequency response H(ejω) =1− e−j2ω

1+ r2e−j2ωwhere r =

0.9. The input to this system is cos nπ/8 + cos nπ/2. The output can be expressed asA1 cos(nπ/8 + θ1) + A2 cos(nπ/2 + θ2). Determine Ai and θi for i = 1, 2.

19. Consider the z-transform pair denoted using the notation x[n]Z←→ X(z), where r1 <

|z| < r2. Let xe[n] be the conjugate symmetric part of x[n], i.e., xe[n] = x∗e [−n]; letxo[n] be the conjugate anti-symmetric part of x[n], i.e., xo[n] = −x∗o [−n]. Similarly,let Xe(z) and Xo(z) be the conjugate symmetric and and anti-symmetric parts of X(z),i.e., Xe(z) = X∗e (z

∗) and Xo(z) = −X∗o (z∗). Important Note: Notationwise, Xe(z) andXo(z) are not the z-transforms of xe[n] and xo[n].

(a) Determine the z-transforms of Rex[n], Imx[n], xe[n], and xo[n].

(b) Specialize the above results for z = ejω, simplifying the result where possible.

(c) Specialize the results in (b) when x[n] is real-valued.

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20. Determine the z-transform for each of the following sequences. Sketch the pole-zeroplot and indicate the RoC. Also indicate whether or not the discrete-time Fourier trans-form exists, and if it does, whether it can be obtained by evaluating the z-transformon the unit circle. (a) δ[n], (b) δ[−n], (c) δ[n − 1], (d) δ[n + 1], (e) −(1/2)nu[−n − 1],(f) (1/2)nu[−n], (g) 1, (h) 7 · (1/3)n cos(2πn/6 + π/4)u[n], (i) 7 · (1/3)n cos(2πn/6 +π/4), (j) an.

21. Let x1[n] = αnu[n] and x1[n] = βnu[n] where 0 < α, β < 1 and α 6= β. Find they[n] = x1[n] ∗ x2[n] by computing the inverse z-transform of X1(z)X2(z). In y[n], ifyou let β −→ α, does it tend to the inverse z-transform of 1/(1− αz−1)2 (with RoC|z| > |a|) ?

22. Consider the finite-duration sequence h[n] defined in the range 0 ≤ n ≤ N − 1. Letg[n] = h∗[N − 1− n].

(a) Express G(z) in terms of H(z).

(b) Now let g[n] = h[n], i.e., h∗[N− 1− n] = h[n]. If z0 is a zero of H(z), can you findanother zero related to z0 ?

(c) Suppose now h[n] is real-valued and z0 is a non-trivial zero that is not locatedon the unit circle. How many other zeros of H(z) can you determine from theknowledge of z0 ?

23. The autocorrelation sequence corresponding to x[n] is denoted by rxx[k] and computedusing the formula x[k] ∗ x∗[−k]. Express the z-transform of rxx[k] in terms of X(z).How are the poles and zeros of the autocorrelation sequence related to those of X(z) ?Ponder about the continuous-time counterpart of this property.

24. Computer assignment Plot the real/imaginary parts, as well as magnitude/phase forvarious values of r and θ of the following DTFT:

G(ejω) =1

1− 2r(cos θ) e−jω + r2 e−j2ω

θ,ω ∈ [0, 2π). In particular, observe what happens when (a) r is close to unity versuswell away, and (b) θ is close to 0 (or π) versus π/2 (or 3π/2). If you consider the

denominator as a polynomial in ejω, where are its roots? Can you relate the peaklocations of the DTFT and the root locations? Although ω is a continuous variable,on a computer you will have to discretize it, which is typically done by taking Nuniformly spaced points in [−π,π).

25. Computer assignment Study the MATLAB function freqz. Consider the G(ejω) givenin the previous problem. Let G = freqz(1, [1, -2*r*cos(theta), r*r], 4000). Plotthe magnitude and phase of G and compare with the result of the previous experiment.Also plot 20*log10(abs(G)) and compare it with it linear-scale counterpart.

26. Computer assignment Study carefully and implement MATLAB Examples 3.1–3.14in the book “DSP Using MATLAB” by Proakis and Ingle. These are important andilluminating examples. ⋄

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