ECS 332: Principles of Communications 2016/1 HW 1 | Due ......ECS 332: Principles of Communications...

ECS 332: Principles of Communications 2016/1

HW 1 — Due: September 2, 5PM

Lecturer: Prapun Suksompong, Ph.D.

Instructions

(a) (1 pt) This assignment has 5 pages. Do not staple or use paper clip. Also, use/printsingle-sided page. Your submitted work will be scanned using automatic documentfeeder.

(b) (2 pt) Write your first name and the last three digit of your student ID on the upper-right corner of every submitted page.

(c) (7 pt) It is important that you try to solve all problems. For each part, write yourexplanation/derivation and answer in the space provided.

(d) Late submission will be heavily penalized.

Problem 1. In class, we have seen how to use the Euler’s formula to show that

cos2x =1

2(cos (2x) + 1) .

For this question, apply similar technique to show that

cosA cosB =1

2(cos (A+B) + cos (A−B)) .

1-1

ECS 332 HW 1 — Due: September 2, 5PM 2016/1

Problem 2. Plot (by hand) the Fourier transforms of the following signals

(a) cos(20πt)

(b) cos(20πt) + cos(40πt)

(c) (cos(20πt))2

(d) cos(20πt)× cos(40πt)

1-2


(e) (cos(20πt))2 × cos(40πt)

Problem 3. Evaluate the following integrals:

(a)

(i)∞∫−∞

2δ (t) dt

(ii)2∫−3

4δ (t− 1) dt

(iii)2∫−3

4δ (t− 3) dt

(b)∞∫−∞

δ (t) e−j2πftdt

(c)

(i)∞∫−∞

δ (t− 2) sin (πt)dt

1-3


(ii)∞∫−∞

δ (t+ 3) e−tdt

(iii)∞∫−∞

e(x−1) cos(π2

(x− 5))δ (x− 3)dx

(d)

(i)∞∫−∞

(t3 + 4) δ (1− t)dt

(ii)∞∫−∞

g (2− t) δ (3− t)dt

(e)2∫−2δ (2t) dt

Problem 4. Consider the signal g(t) shown in Figure 1.1.

1

t

0.5

1

612 15 24

𝒈 𝒕

Figure 1.1: Problem 4

(a) Carefully sketch the following signals:

(i) y1(t) = g(−t)(ii) y2(t) = g(t+ 6)

1-4


(iii) y3(t) = g(3t)

(iv) y4(t) = g(6− t).

(b) Find the area under the curve (integrate from −∞ to +∞) for each of the signals inthe previous part.

1-5

Q1 Euler's FormulaThursday, November 11, 2010 2:54 PM

ECS332_2016_HW_1_Sol Page 1

Q2 Fourier transforms of cosine functionsWednesday, August 19, 2015 5:07 PM


Q3 Integrals involving delta functionWednesday, July 06, 2011 12:46 PM


All the signals are plotted below

Q4 Time manipulationWednesday, July 06, 2011 12:20 PM


All the signals are plotted below





Instructions

(a) (1 pt) This assignment has 7 pages. Use/print single-sided page. Your submitted workwill be scanned using automatic document feeder.


(c) (7 pt) It is important that you try to solve all problems. For each part, write yourexplanation/derivation and answer in the space provided. The extra question at theend is optional.

Problem 1.

(a) Plot (by hand) the amplitude spectrum of the signal x(t) =

{1, −4 < t < 4,0, otherwise.

(b) Plot (by hand) the amplitude spectrum of the signal x(t) =

{2, −2 < t < 2,0, otherwise.

2-1


(c) Plot (by hand) the amplitude spectrum of the signal x(t) =

{2, −3 < t < 1,0, otherwise.

Problem 2. 1 Using MATLAB to find the (amplitude) spectrum2 of a signal:A signal g(t) can often be expressed in analytical form as a function of time t, and the

Fourier transform is defined as the integral of g(t) exp(−j2πft). Often however, there is noanalytical expression for a signal, that is, there is no (known) equation that represents thevalue of the signal over time. Instead, the signal is defined by measurements of some physicalprocess. For instance, the signal might be the waveform at the input to the receiver, theoutput of a linear filter, or a sound waveform encoded as an mp3 file.

In all these cases, it is not possible to find the spectrum by analytically performing aFourier transform. Rather, the discrete Fourier transform (or DFT, and its cousin, the morerapidly computable fast Fourier transform, or FFT) can be used to find the spectrum orfrequency content of a measured signal. The MATLAB function plotspect.m, which plots thespectrum of a signal can be downloaded from our course website. Its help portion3 notes

% plotspect(x,t) plots the spectrum of the signal x% whose values are sampled at time (in seconds) specified in t

(a) The function plotspect.m should be straightforward to use. For instance, the spec-trum of a rectangular pulse4 g(t) = 1[0 ≤ t ≤ 2] can be found using:

1Based on [Johnson, Sethares, and Klein, 2011, Sec 3.1 and Q3.3].2also referred to by “amplitude spectrum” or simply “spectrum”3You can view the “help” portion for a MATLAB function xxx by typing help xxx at the MATLAB prompt.

If you get an error such as xxx not found, then this means either that the function does not exist, or that itneeds to be moved into MATLAB’s search path.

4Here, we define a rectangular pulse using the indicator function 1[·]. This function outputs a 1 when thestatement inside the square brackets is true; otherwise, it outputs a 0. For example,

1 [0 ≤ t ≤ 2] =

{1, 0 ≤ t ≤ 2,0, otherwise.

2-2


% specrect.m plot the spectrum of a square waveclose allTs=1/100; % time interval between adjacent samplest=0:Ts:20; % create a time vectorx=[t ≤ 2]; % rectangular pulse 1[0 ≤ t ≤ 2]plotspect(x,t) % call plotspect to draw spectrumxlim([−5,5]) % look only from f = −5 to f = 5 Hz

The output of specrect.m is shown in Figure 2.1. The top plot shows the first 20seconds of g(t). The bottom plot shows |G(f)|.

0 2 4 6 8 10 12 14 16 18 200

0.5

1

Seconds

−5 −4 −3 −2 −1 0 1 2 3 4 50

1

2

3

Frequency [Hz]

Mag

nitu

de

Figure 2.1: Plots from specrect.m

(i) Use what we studies in class about the Fourier transform of a rectangular pulse(and the time-shift property) to find a simplified expression for |G(f)|.

(ii) Use MATLAB to plot your analytical expression derived in part (i). Did your plotagree with the lower plot in Figure 2.1? Attach the printed plot on another page;write the page number as page 2-8. (Remark: There will be more plots to be puton this page.)

2-3


(b) Now consider an exponential pulse

s(t) = e−tu(t).

(i) Modify the code in specrect.m to show the (magnitude) spectrum |S(f)|. Includethe printed plot on page 2-8.

(ii) Find S(f) and |S(f)| analytically. (Hopefully, you still remember how to integrateexponential function.)

(iii) Plot your analytical expression in part (ii) and compare with the plot in part (i).Include the printed plot on page 2-8.

(iv) MATLAB can also perform symbolic manipulation when symbolic toolbox is in-stalled. It can find the Fourier transform of a symbolic expression via the com-mand fourier. Unfortunately, the fourier command use the ω-version of thedefiition. So, to convert the answer to the f -version, we also need to substituteω = 2πf . This is done automatically in our provided function fourierf.

Run the file SymbFourier.m. Check whether you have the same result as part(ii).

2-4


Problem 3.

(a) Suppose the Fourier transform of a signal x(t) is given by

X (f) = sinc (5πf) =sin (5πf)

(5πf).

(i) Plot (by hand) x(t).

(ii) Find∫∞−∞X(f)df . (Hint: This integration is exactly the inverse Fourier transform

formula with t = 0.)

(b) Suppose the Fourier transform of a signal y(t) is given by

Y (f) = sinc2 (5πf) =

(sin (5πf)

(5πf)

)2

.

(i) Plot (by hand) y(t).

(ii) Find∫∞−∞ Y (f)df .

2-5


Problem 4. The Fourier transform of the triangular pulse g(t) in Figure 2.2a is given as

G(f) =1

(2πf)2(ej2πf − j2πfej2πf − 1

)Using this information, and the time-shifting and time-scaling properties, find the Fouriertransforms of the signals shown in Figure 2.2b, c, d, e, and f.

0

1

‐1t

g(t)

0

1

1t

g1(t)

(a) (b) (c)

0

1

1t

g2(t)

2

‐1

1

0t

g3(t)

1

(d) (e) (f)

1

1

1t

g4(t)

2 2‐ 0 0

1.5

t

g5(t)

2

Figure 2.2: Problem 4

Remark: Don’t forget to simplify your answers. For example, the answer in part (d)should be of the form sinc2(·) and the answer in part (e) should be of the form sinc(·)

2-6


Extra Question

Here is an optional question for those who want more practice.

Problem 5. Listen to the Fourier’s Song (Fouriers Song.mp3) which can be downloadedfrom

http://sethares.engr.wisc.edu/mp3s/fourier.html

Which properties of the Fourier Transform can you recognize from the song? List them here.

Don’t forget to write your first name and the last three digit of your student ID on theupper-right corner of every submitted page.

2-7

Q1 Rectangular pulses and sinc functionsWednesday, August 26, 2015 8:36 PM

ECS332 2016 HW2 Page 1

Q2 Magnitude Spectrum via MATLAB

a)

i) The plot in the time domain shows a rectangular pulse g t on the interval [0, 2].

In class, we have seen how to find the Fourier transform of rectangular functions

that are even functions (symmetric wrt. the vertical axis). Our g t is not. We need

to shift it to the left by 1 to get a rectangular pulse 1 1t which is even.

As we have discussed in class, time shifting does not change the amplitude

spectrum. Hence, G f is the same as the magnitude of the Fourier transform of

1 1t .

Method 1:

Method 2:

In class, we have seen an example of a Fourier transform pair where we have

rectangular pulse of width T0 centered at origin in the time domain. In particular, we

know that

0

0 01 sinc2

Tt T T f

.

Pluggging in the value of 2 for the width, we have

1 1 2sinc 2t f .

Therefore,

2 sinc 2G f f .

0 2 4 6 8 10 12 14 16 18 200

0.5

1

Seconds

-5 -4 -3 -2 -1 0 1 2 3 4 50

1

2

3

Frequency [Hz]

Magnitude

ii) In the bottom part of Figure (i) below, the theoretical expression in part (i) is plotted

using the “x” marks on top of the provided plot from specrect.m. The marks

match the theoretical plot. Therefore, the expression above agrees with the plot via

MATLAB’s plotspect.m.

Figure (i) Figure (ii)

b)

i) See Figure (ii) above.

ii) By the Fourier-transform formula,

1 22 2

0

1 2

0

1

1

1

1 22

j f tj ft t j ft

j f t

t

S

j f

f s t e dt e u t e dt e dt

ej f

Recall that the magnitude of a complex number z x jy is 2 2z x y and that

11

2 2

zz

z z .

Therefore,

2

1

1 2 f

S f

.

iii) The S f derived analytically is plotted in Figure (ii) using the “x” marks on top of

the plots from plotspect.m. They are virtually identical.

iv) Here is the result displayed on the command window:

>> SymbFourier

S =

1/(a + pi*f*2*i)

With variable “a” in the m-file set to 1, we have same result as in (i).

0 2 4 6 8 10 12 14 16 18 200

0.5

1

Seconds

-5 -4 -3 -2 -1 0 1 2 3 4 50

1

2

3

Frequency [Hz]

Magnitude

Use plotspect

Theoretical

0 2 4 6 8 10 12 14 16 18 200

0.5

1

Seconds

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.5

1

1.5

Frequency [Hz]

Magnitude

Use plotspect

Theoretical

Q3 Sinc Function and Triangular SignalWednesday, July 06, 2011 12:16 PM


Q4 Using Properties of FTThursday, August 27, 2015 8:54 PM





Instructions

(a) This assignment has 5 pages.


(c) (1 pt) For each part, write your explanation/derivation and answer in the space pro-vided.

(d) (8 pt) It is important that you try to solve all non-optional problems.

(e) Late submission will be heavily penalized.

Problem 1 (M2011). The Fourier transform X(f) for a signal x(t) is shown in Figure 3.1.

-3 5

8

f

X f

Figure 3.1: Plot of X(f) for Problem 1.

Let g(t) = x(−2t) and y (t) = x (4− 2t). Carefully sketch |G(f)| and |Y (f)|.

3-1


Problem 2. 1

(a) Consider the cosine pulse

p (t) =

{cos (10πt) , −1 ≤ t ≤ 10, otherwise

(i) Sketch p(t) for −3 ≤ t ≤ 3.

(ii) Find P (f) analytically.

(iii) Sketch P (f) from -10 Hz to 10 Hz.

1Inspired by [Carlson and Crilly, 2009, Q2.2-1 and Q2.2-2].

3-2


(b) Consider the cosine pulse

p (t) =

{cos (10πt) , 2 ≤ t ≤ 40, otherwise

(i) Find P (f) analytically.

(ii) Use MATLAB. Mimic the code in specrect.m to plot the spectrum of p(t). Followthe settings below:

• Consider the time t from 0 to 10 [s] when you set up the time vector.

• Use the sampling frequency of 500 samples per sec. So, the sampling interval(the time between adjacent samples) is Ts = 1/500.

• With the above sampling frequency, plotspect will plot the magnitude spec-trum from -250 to 250 Hz. Use the function xlim (or the magnifier glass GUI)to limit your frequency view to be only from -10 to +10 Hz.

(iii) Also in MATLAB, add the plot of your analytical answer from part (i) into the samefigure as part (ii).

• Print this figure and attach it at the end of your HW.

• On this attached page, compare the two plots. (Write some description/ob-servation. Are they the same? How can you tell?)

Caution: The built-in sinc function in MATLAB is defined using the normalizedversion. So, you will need to remove a factor of π from the argument of each sincfunction found in part (i) when you type it into MATLAB.

3-3


Problem 3. You are given the baseband signals (i)m(t) = cos 1000πt; (ii)m(t) = 2 cos 1000πt+cos 2000πt; (iii) m(t) = (cos 1000πt)× (cos 3000πt). For each one, do the following.

(a) Sketch the spectrum of m(t).

(b) Sketch the spectrum of the DSB-SC signal m(t) cos 10, 000πt.

[Lathi and Ding, 2009, Q4.2-1]

3-4


Problem 4 (M2011). Use properties of Fourier transform to evaluate the following integrals.

(Do not integrate directly. Recall that sinc(x) = sin(x)x

.) Clearly state the property orproperties that you use.

(a)∞∫−∞

sinc(√

5x)dx

(b)∞∫−∞

sinc(√

5x)

sinc(√

7x)dx

(c) (Optional)∞∫−∞

e−2πf×2j2sinc (2πf)(e−2πf×5j2sinc (2πf)

)∗df

(d) (Optional)∞∫−∞

sinc (π (x− 5)) sinc(π(x− 7

2

))dx

3-5

Q1 Time Manipulation and Fourier TransformWednesday, September 2, 2015 5:36 PM


Q2 Cosine PulsesWednesday, July 18, 2012 3:43 PM


Q3 Tone ModulationThursday, July 14, 2011 4:40 PM


Q4 Integrations involving sinc function(s)Thursday, January 22, 2015 6:38 PM





Instructions

(a) This assignment has 8 pages.


(c) (1 pt) For each part, write your explanation/derivation and answer in the space pro-vided.

(d) (8 pt) It is important that you try to solve all problems.

(e) Late submission will be heavily penalized.

Problem 1. Given a system with input-output relationship of

y(t) = 2x(t) + 10,

is this system linear? [Carlson and Crilly, 2009, Q2.3-10]

Problem 2. Signal x(t) = 10 cos(2π × 7× 106 × t) is transmitted to some destination. Thereceived signal is y(t) = 10 cos(2π × 7× 106 × t− π/6).

(a) What is the minimum distance between the source and destination?

4-1


(b) What are the other possible distances?

[Carlson and Crilly, 2009, Q2.3-14]

Problem 3. Consider the DSB-SC modem with no channel impairment shown in Figure3.1. Suppose that the message is band-limited to B = 3 kHz and that fc = 100 kHz.

× ×Channel

3cos 2 cf t

y

cos 2 cf t

vLPF

Modulator Demodulator

Message(modulating signal)

Figure 4.1: DSB-SC modem with no channel impairment

(a) Specify the frequency response HLP (f) of the LPF so that m̂(t) = m(t).

4-2


(b) Suppose the transfer function hLP (t) of the LPF is of the form α sinc(βt). Find theconstants α and β such that m̂(t) = m(t).

Problem 4. This question starts with a square-modulator for DSB-SC . Then, the use ofthe square-operation block is further explored on the receiver side of the system. [Doerschuk,2008, Cornell ECE 320]

(a) Let x(t) = Acm(t) where m(t)F−−⇀↽−−F−1

M(f) is bandlimited to B, i.e., |M(f)| = 0 for

|f | > B. Consider the block diagram shown in Figure 3.2.

x t + u t

2 cos 2 cf t

2

BPH f v t y t

Figure 4.2: Block diagram for Problem 4a

Assume fc � B and

HBP (f) =

1, |f − fc| ≤ B1, |f + fc| ≤ B0, otherwise.

4-3


The block labeled “{·}2” has output v(t) that is the square of its input u(t):

v(t) = u2(t).

Find y(t).

(b) The block diagram in part (a) provides a nice implementation of a modulator becauseit may be easier to build a squarer than to build a multiplier. Based on the successfuluse of a squaring operation in the modulator, we decide to use the same squaringoperation in the demodulator. Let

x (t) = Acm (t)√

2 cos (2πfct)

where m(t)F−−⇀↽−−F−1

M(f) is bandlimited to B, i.e., |M(f)| = 0 for |f | > B. Again,

assume fc � B Consider the block diagram shown in Figure 3.3.

x t +

2 cos 2 cf t

2

LPH f Iy t

x t +

2 sin 2 cf t

2

LPH f Qy t

Figure 4.3: Block diagram for Problem 4b

Use

HLP (f) =

{1, |f | ≤ B0, otherwise.

4-4


Find yI(t). Does this block diagram work as a demodulator; that is, is yI(t) propor-tional to m(t)?

(c) Due to the failure in part (b), we have to think hard and it seems natural to consideralso the block diagram with cos replaced by sin. Let

x (t) = Acm (t)√

2 cos (2πfct)

where m(t)F−−⇀↽−−F−1

M(f) is bandlimited to B, i.e., |M(f)| = 0 for |f | > B as in part (b).

Again, assume fc � B Consider the block diagram shown in Figure 3.4.

x t +

2 cos 2 cf t

2

LPH f Iy t

x t +

2 sin 2 cf t

2

LPH f Qy t

Figure 4.4: Block diagram for Problem 4c

4-5


As in part (b), use

HLP (f) =


Find yQ(t).

(d) Use the results from parts (b) and (c). Draw a block diagram of a successful DSB-SCdemodulator using squaring operations instead of multipliers.

4-6


Problem 5 (Cube modulator). Consider the block diagram shown in Figure 3.5 where“{·}3” indicates a device whose output is the cube of its input.

m t +

02 cos 2 f t

3

H f z t x t y t

Figure 4.5: Block diagram for Problem 5. Note the use of f0 instead of fc.

Let m(t)F−−⇀↽−−F−1

M(f) be bandlimited to B, i.e., |M(f)| = 0 for |f | > B.

(a) Plot an H(f) that gives z (t) = m (t)√

2 cos (2πfct). What is the gain in H(f)? Whatis the value of fc? Notice that the frequency of the cosine is f0 not fc. You are supposedto determine fc in terms of f0.

(b) Let M(f) be

M (f) =


4-7


(i) Plot X(f).

(ii) Plot Y (f). Hint:

M (f) ∗M (f) =

{2B − |f | , |f | ≤ 2B0, otherwise.

Do not attempt to make an accurate plot or calculation for the Fourier transformof m3(t).

(iii) For your filter of part (a), plot z(t).

[Doerschuk, 2008, Cornell ECE 320]

4-8

Q1 Linear SystemWednesday, July 18, 2012 4:57 PM


Q2 Time DelayWednesday, July 18, 2012 5:17 PM


Q3 DSB-SCTuesday, September 15, 2015 12:28 PM


Q4 Square MODEMThursday, July 14, 2011 2:22 PM


Q5 Cube ModulatorThursday, July 14, 2011 2:11 PM



HW 5 — Not Due


Problem 1. Consider the two signals s1(t) and s2(t) shown in Figure 5.1. Note that V andTb are some positive constants. Your answers should be given in terms of them.

178 Optimum receiver for binary data transmission�

−V

t

V

0

Tb

t

Tb1

0

Tb

t

V

0 Tb

Tb−1

t0 Tb

Tb1

(a)

(b)

φ1(t) φ2(t)

s2(t)s1(t)

�Fig. 5.5 (a) Signal set for Example 5.2, (b) orthonormal functions.

0

E

E

φ2(t)

φ1(t)

s2(t)

s1(t)�Fig. 5.6 Signal space representation for Example 5.2.

Graphically, the orthonormal basis functions φ1(t) and φ2(t) look as in Figure 5.5(b) andthe signal space is plotted in Figure 5.6. The distance between the two signals can be easilycomputed as follows:

d21 =√

E + E = √2E = √2√

E. (5.35)

�

In comparing Examples 5.1 and 5.2 we observe that the energy per bit at the transmitteror sending end is the same in each example. The signals in Example 5.2, however, are closertogether and therefore at the receiving end, in the presence of noise, we would expect moredifficulty in distinguishing which signal was sent. We shall see presently that this is thecase and quantitatively express this increased difficulty.

Example 5.3 This is a generalization of Examples 5.1 and 5.2. It is included princi-pally to illustrate the geometrical representation of two signals. The signal set is shown

Figure 5.1: Signal set for Question 1

(a) Find the energy in each signal.

(b) Are they energy signals?

(c) Are they power signals?

5-1

ECS 332 HW 5 — Not Due 2016/1

(d) Find the (average) power in each signal.

(e) Are the two signals s1(t) and s2(t) orthogonal?

Problem 2. (Power Calculation) For each of the following signals g(t), find (i) its corre-sponding power Pg =

⟨|g (t)|2

⟩, (ii) the power Px =

⟨|x (t)|2

⟩of x (t) = g (t) cos (10t), and

(iii) the power Py =⟨|y (t)|2

⟩of y (t) = g (t) cos (50t)

(a) g (t) = 3 cos (10t+ 30◦).

5-2

ECS 332 HW 5 — Not Due 2016/1

(b) g (t) = 3 cos (10t+ 30◦) + 4 cos (10t+ 120◦). (Hint: First, use phasor form to combinethe two components into one sinusoid.)

(c) g (t) = 3 cos (10t) + 3 cos (10t+ 120◦) + 3 cos (10t+ 240◦)

5-3

ECS 332 HW 5 — Not Due 2016/1

Problem 3. Consider a signal g(t). Recall that |G(f)|2 is called the energy spectraldensity of g(t). Integrating the energy spectral density over all frequency gives the signal’stotal energy. Furthermore, the energy contained in the frequency band I can be found fromthe integral

∫I|G(f)|2df where the integration is over the frequencies in band I. In particular,

if the band is simply an interval of frequency from f1 to f2, then the energy contained inthis band is given by ∫ f2

f1

|G(f)|2df. (5.1)

In this problem, assumeg(t) = 1[−1 ≤ t ≤ 1].

(a) Find the (total) energy of g(t).

(b) Figure 5.2 define the main lobe of a sinc pulse. It is well-known that the main lobeof the sinc function contains about 90% of its total energy. Check this fact by firstcomputing the energy contained in the frequency band occupied by the main lobe andthen compare with your answer from part (a).

Hint: Find the zeros of the main lope. This give f1 and f2. Now, we can apply (5.1).MATLAB or similar tools can then be used to numerically evaluate the integral.

5-4

ECS 332 HW 5 — Not Due 2016/1sinc function

1

Main lobe(null to null)

Figure 5.2: Main lobe of a sinc pulse

(c) Suppose we want to include more energy by considering wider frequency band. Letthis band be the interval I = [−f0, f0]. Find the minimum value of f0 that allows theband to capture at least 99% of the total energy in g(t).

Problem 4. Consider a “square” wave (a train of rectangular pulses) shown in Figure 5.3.Its values periodically alternates between two values A and 0 with period T0. At t = 0, itsvalue is A.

1

A

… …

Width

Figure 5.3: A train of rectangular pulses

Some values of its Fourier series coefficients are provided in the table below:

5-5

ECS 332 HW 5 — Not Due 2016/1

k -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

ck −√2

7π− 1

3π−√2

5π0

√2

3π1π

√2π

12

√2π

1π

√2

3π0 −

√2

5π− 1

3π−√2

7π

(a) Find its duty cycle.

(b) Find the value of A. (Hint: Use c0.)

Extra Question

Here is an optional question for those who want more practice.

Problem 5 (M2011). In this question, you are provided with a partial proof of an importantresult in the study of Fourier transform. Your task is to figure out the quantities/expressionsinside the boxes labeled a,b,c, and d.

We start with a function g(t). Then, we define x (t) =∞∑

`=−∞g (t− `T ). It is a sum that

involves g(t). What you will see next is our attempt to find another expression for x(t) interms of a sum that involves G(f).

5-6

ECS 332 HW 5 — Not Due 2016/1

To do this, we first write x(t) as x (t) = g (t)∗∞∑

`=−∞δ (t− `T ). Then, by the convolution-

in-time property, we know that X(f) is given by

X (f) = G (f)× a∞∑

`=−∞

δ(f + b

)

We can get x(t) back fromX(f) by the inverse Fourier transform formula: x (t) =∞∫−∞

X (f) ej2πftdf .

After plugging in the expression for X(f) from above, we get

x (t) =

∞∫−∞

ej2πftG (f) a∞∑

`=−∞

δ(f + b

)df

= a

∞∫−∞

∞∑`=−∞

ej2πftG (f) δ(f + b

)df.

By interchanging the order of summation and integration, we have

x (t) = a∞∑

`=−∞

∞∫−∞

ej2πftG (f) δ(f + b

)df.

We can now evaluate the integral via the sifting property of the delta function and get

x (t) = a∞∑

`=−∞

e c G(

d).

5-7

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ECS 332: Principles of Communications 2016/1 HW 1 | Due ......ECS 332: Principles of Communications...

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