ENEE 420/FALL 2010
COMMUNICATION SYSTEMS HOMEWORK # 1: Please work out the ten (10) problems stated below – LD refers to the text: B,P. Lathi and Z. Ding, Modern Digital and Analog Communication Systems (Fourth Edition), Oxford University Press, Oxford (UK), 2009. Exercise 2.1-1 (LD) refers to Exercise 1 for Section 2.1 of LD. Show work and explain reasoning. Three (3) problems, selected at random amongst these ten problems, will be marked. 1. Problem 2.1-2 (LD). 2. Problem 2.1-3 (LD). 3. Problems 2.2-1 and 2.2-2 (LD). 4. Problem 2.3-4 (LD). 5. Problem 2.4-1 (LD). 6. Problem 2.5-5 (LD). 7. Problem 2.8-4 (LD). 8. Problem 2.9-1 (LD). 9. Problem 2.9-3 (LD).
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COMMUNICATION SYSTEMS HOMEWORK # 2: Please work out the ten (10) problems stated below – LD refers to the text: B,P. Lathi and Z. Ding, Modern Digital and Analog Communication Systems (Fourth Edition), Oxford University Press, Oxford (UK), 2009. Exercise 2.1-1 (LD) refers to Exercise 1 for Section 2.1 of LD. Show work and explain reasoning. Three (3) problems, selected at random amongst these ten problems, will be marked. 1. Problem 3.1-1 (LD). 2. Problem 3.1-2 (LD). 3. Problems 3.1-4 (LD). 4. Problems 3.1-5 (LD). 5. Problems 3.1-6 (LD). 6. Problems 3.1-7 (LD). 7. Problems 3.3-2 (LD). 8. Problems 3.3-3 (LD): Do the calculations in two different ways: (i) As requested in the textbook and (ii) By direct calculations! 9. Problems 3.4-1 (LD):
ENEE 420/FALL 2010 10. Consider a periodic signal g : R → C with period T > 0, i.e., g(t + T ) = g(t), Assume that Z
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COMMUNICATION SYSTEMS HOMEWORK # 2 – An Answer to Ex. 10 Consider a periodic signal g : R → C with period T > 0, i.e., t ∈ R.
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For any scalar a in R, it is always possible to write a = `T + α for some ` = 0, ±1, . . . and 0 ≤ α < T – Note that ` and α are uniquely determined by a once T > 0 is given. Now, for any mapping h : R → C which is periodic with period T , whenever the integrability condition Z T
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ENEE 420/FALL 2010 Indeed, we have Z a+T Z T h(t)dt = h(a + s)ds [Change of variable: t = a + s] a 0 Z T = h(`T + α + s)ds [Use the fact that a = `T + α] 0 Z T h(α + s)ds [Use the fact that h is periodic of period T ] = 0 Z α Z T = h(α + s)ds + h(α + s)ds 0 α Z α Z T −α = h(α + s)ds + h(x)dx [Change of variable x = α + s] 0 0 Z T Z T −α = h(y)dy + h(x)dx [Change of variable y = T − α + s] T −α 0 Z T = h(x)dx (1.2) 0
and (1.1) is established. The exercise is solved by applying (1.1) to the functions t∈R n = 0, ±1, 2, . . .
n
hn (t) = g(t)e−j2π T t ,
where g : R → C is periodic with period T and satisfies the integrability condition Z T |g(t)|dt < ∞. 0
Indeed, for each n = 0, ±1, ±2, . . ., the function hn : R → C is periodic of period T since hn (t + T ) = = = = =
n
g(t + T )e−j2π T (t+T ) n g(t)e−j2π T (t+T ) [Recall that g is periodic with period T ] n g(t)e−j2πn e−j2π T t n g(t)e−j2π T t hn (t), n = 0, ±1, ±2, . . .
Also Z
T
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0
and the integrability condition is satisfied.
Z 0
T
|g(t)|dt < ∞
(1.3)
ENEE 420/FALL 2010
COMMUNICATION SYSTEMS HOMEWORK # 3: Please work out the ten (10) problems stated below – LD refers to the text: B,P. Lathi and Z. Ding, Modern Digital and Analog Communication Systems (Fourth Edition), Oxford University Press, Oxford (UK), 2009. Exercise 2.1-1 (LD) refers to Exercise 1 for Section 2.1 of LD. Show work and explain reasoning. Three (3) problems, selected at random amongst these ten problems, will be marked. 1. Problem 3.4-3 (LD). Do the calculations in two different ways: (i) by using properties of Fourier transforms and (ii) by direct calculations. 2. Problem 3.5-1 (LD). 3. Problem 3.5-2 (LD). 4. Problems 3.5-3 (LD). 5. Problem 3.6-1 (LD). 6. In class we use the following approximation trick to find the Fourier transform of the signum function sgn : R → R given by −1 if t < 0 1 if t = 0 sgn(t) = 2 1 if t > 0 We do so by considering the approximating functions ga : R → R given by ga (t) = −u(−t)eat + u(t)e−at ,
t∈R
ENEE 420/FALL 2010 with a > 0. We then computed the Fourier transform Ga (f ) of ga , and took the limit with a ↓ 0. This gave us the Fourier transform pairing sgn(t) ⇐⇒
1 jπf
Here we are going to follow a similar approach but with different approximations. Consider approximating the signum function sgn : R → R by the functions hT : R → R given by 0 if t < −T −1 if −T ≤ t < 0 0 if t = 0 hT (t) := 1 if 0 < t ≤ T 0 if T < t
with T > 0. Compute the Fourier transform HT (f ) of hT . What happens to HT (f ) when T goes to infinity so that hT becomes a better and better approximation to the signum function sgn : R → R? Any comment or insight? 7. Parts (a)-(c) of Problem 3.6-3 (LD). 8. Problem 3.7-1 (LD) – Hint: Z R
for any function g : R → C. 9. Problem 3.7-4 (LD). 10. Problem 3.7-5 (LD).
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ENEE 420/FALL 2010
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ENEE 420/FALL 2010
COMMUNICATION SYSTEMS HOMEWORK # 5: Please work out the ten (10) problems stated below – LD refers to the text: B,P. Lathi and Z. Ding, Modern Digital and Analog Communication Systems (Fourth Edition), Oxford University Press, Oxford (UK), 2009. Exercise 2.1-1 (LD) refers to Exercise 1 for Section 2.1 of LD. Show work and explain reasoning. Three (3) problems, selected at random amongst these ten problems, will be marked. 1. Problem 4.3-7 (LD). 2. Problem 4.3-8 (LD). 3. Problem 4.3-9 (LD). 4. Problem 4.4-1 (LD). 5. Consider a signal g : R → R such that Z |g(t)|dt < ∞ and R
Z
|g(t)|2 dt < ∞.
R
Explain how to make sense of the following statement: The Hilbert transform gh : R → R of g is well defined whenever we have Z |G(f )|df < ∞. R
and in the process give an expression for gh . Compute the energy Z |gh (t)|2 dt, R
ENEE 420/FALL 2010 and show that g and gh are orthogonal signals in the sense that Z g(t)gh (t)dt = 0. R
6. Problem 4.4-3 (LD). 7. Consider the quadrature filter with frequency −j π e 2 0 H(f ) = j π2 e
response if f > 0 if f = 0 if f < 0 .
This filter can used to implement the Hilbert transform. We can generalize this concept to a new transform which introduces a phase shift of θ in the frequency component. Thus, with −jθ e if f > 0 0 if f = 0 Hθ (f ) = jθ e if f < 0 . we introduce the θ-transform of the signal g : R → R as the signal gθ : R → R determined through Gθ (f ) = Hθ (f )G(f ), f ∈ R. As in Problem 5, give conditions to ensure that gθ is well defined. Under these conditions show that gθ is a linear combination of g and of its Hilbert transform gh . 8. Problem 4.4-5 (LD). 9. Problem 4.4-6 (LD). 10. Problem 4.4-7 (LD).
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COMMUNICATION SYSTEMS HOMEWORK # 6: Please work out the ten (10) problems stated below – LD refers to the text: B,P. Lathi and Z. Ding, Modern Digital and Analog Communication Systems (Fourth Edition), Oxford University Press, Oxford (UK), 2009. Exercise 2.1-1 (LD) refers to Exercise 1 for Section 2.1 of LD. Show work and explain reasoning. Three (3) problems, selected at random amongst these ten problems, will be marked. 1. Problem 5.1-1 (LD). 2. Problem 5.1-2 (LD). 3. Problem 5.1-3 (LD). 4. Problem 5.1-4 (LD). 5. Problem 5.1-5 (LD). 6. Problem 5.2-1 (LD). 7. Problem 5.2-2 (LD). 8. Problem 5.2-4 (LD). 9. Show that Jk (β) is real for all k = 0, ±1, . . . and β in R. 10. Draw the block diagram for the FDM system with twelve voice channels as depicted in Figure 4.25 (p. 212). Identify the blocks for both modulation and demodulation.
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