(a) Use Tables of discrete-time Fourier transform to find the frequency spectra of the sig...

(a) Use Tables of discrete-time Fourier transform to find the frequency spectra of the signals listed subsequently.

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Table Discrete-Time Fourier Transforms

f[n]	F(Ω)
1. δ[n]	1
2. 1
3. u[n]
4. an u[n]; |a|<1
5. nan u[n]; |a|<1
6. a–nu[–n – 1]; |a|<1
7.
8. cos[Ω0n]
9. sin[Ω0n]
10. x[n] periodic with period N

Table Frequency Response Data for Example 12.4

Ω (rad/sample)	ω = Ω (rad/s)	|H(Ω)|	∠H(Ω)
0	0.0	1	0°
0.976	97.6	0.7071	–55.9°
	104.7	0.66	–60°
	157.1	0.333	–90°
	209.4	0.0	–120°
2.80	280.0	2.948	–160.4°
π	314.2	0.333	–180°

Table Properties of the Discrete-TimeFourier Transform

Signal	Transform
x[n]
x[n]	X(Ω) = X(Ω + 2π)
a1x1[n] + a2x2[n]	a1X1(Ω) = a2X2(Ω)
x[n – n0]
	X(Ω – Ω0)
x[n] real
x[–n]	X(–Ω)
x[n]*y[n]	X(Ω)Y(Ω)
x[n]y[n]
nx[n]

(i) f1(t) = 8 cos(2πt) + 4 sin(4πt), sampled with Ts = 0.1 s.

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(ii) f2[n] = 5 cos [0.5πn]a[n].

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(iii) f3(t)= 2 sin(3πt) + 3 cos(5πt), T = 0.1 s.

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(iv) f4[n] = 3 cos [0.6πn].

(b) Plot the magnitude and phase frequency spectra of each of the signals listed over the frequency range | ω | ≦ 2 ωs