Assignm

ent #1 -- Oct.21, 2020

1. [5 mark

s] We consider raised cosine (R

C) pulse shaped transm

ission. The im

pulse response of the pulse-shaping continuous time filter g(t) is given

as follows:

g(t) = S

inc( t/T) { C

os(t/T

) / (1-42t 2/T

2) }

where the excess bandw

idth factor 0 ≤≤

1 and T is the sym

bol duration and Sinc(x)=

Sin(x)/x. W

e use a discrete-time filter g(nT

s) = g(n) w

here the sam

pling rate Ts =

T/J, w

here J is a positive integer. Also, the filter is truncated to 2L

T sym

bol durations, i.e., LT

symboldurations on each side of t=

0, w

here L is a positive integer. T

he input bit (or symbol) stream

I(kT) into this filter w

ill be zero-interleaved with J zeros in order to create the output

sequence x(kTs) w

hich is then sent into a DA

C.

Chose your ow

n 16-bit random sequence w

here I(k){+

1, -1} in order to create a BP

SK

signal. Assum

e all the bits before and after this 16-bit pattern are zeros. Y

ou are to plot the output sequence (samples) x(kT

s) using both “symbol-plot” as w

ell as join them using “line-plot” using M

atlab.

(a) Plot as F

ig.1, x(kTs) for J=

4, L=

2, and =

0.(b) R

epeat (a) and include in Fig.1 another plot with the change L

=4.

(c) Plot as Fig.2, three

different plots of x(kTs) w

here all of them have J=

8, L=2, but w

ith threedifferent excess bandw

idth factors, namely,

=0,

=0.5,

and =

1.(d) C

omm

ent on your results in (b) and (c).

2. [15 mark

s]In this question, w

e compare the theoretical probability of sym

bol error P(e) =

PS

and/or probability of bit error, PB , w

ith computer sim

ulated sym

bol or bit error rate (SE

R or B

ER

), for a few popular linear m

odulation schemes. T

he approximation on P

Susing bounds w

ill also be studied.

(a) For the same energy per bit, E

b, compute and plot on the sam

e figure (Fig-1) the PS

of BP

SK

, QP

SK

, and 16-QA

M signals. A

ssume the com

plex base-band A

WG

N m

easurement m

odel r(k)=I(k)+

v(k), where iid I(k) is uncorrelated w

ith the white G

aussian noise v(k). For complex signals

(QP

SK

and 16-QA

M) the noise

will be com

plex with variance N

O /2 per dimension. V

ary the ratio 10log10 (E

b/No) in 2dB

steps from 0dB

to 10dB, by changing the noise variance and plot against

log10 (P

S ), for all the 3 signals. Chose E

b=1 for all the signals. N

ote that the log of the error probability is plotted on the y-axis. The Q

(.) function or the erfc(.) function can be evaluated using M

atlab.

(b) Assum

ing Gray coding, plot 10log

10 (Eb/N

o) versus log10 (P

B ) for the 3 signals. Add these curves also to F

ig-1, and label neatly.

(c) For the QP

SK

signal set, compute using the follow

ing approximations to the sym

bol error probability. Plot each of these values of P

S for the range of SN

Rs

from 0dB

to 16dB in the sam

e figure. Call it Fig-2.

(c1) Union bound using allthe pairw

ise symbol errors

(c2) Union bound using only the nearest neighbours

(c3) In part (c2) replace the erfc() function with the C

hernoff bound and add this also to Fig-2.

(d) Now

, we w

ish to simulate the sym

bol error rate (SE

R) for Q

PS

K. For generating the transm

it symbols, use uniform

random variable (rv) betw

een (0,1). If the rv takes a value X

between 0 and 0.5, it is m

apped to say -d; else, it is mapped to +

d, where 2d is the distance betw

een the symbols. W

e will need 2 rvs, one of the

real part and one for the imaginary part. C

hose “d” to ensure Eb=

1. Generate 10

5sym

bols to measure the S

ER

over the same range of SN

Rs as in part (c). P

lot your results back in Fig-2. C

omm

ent.

(e) Repeat part (c) above for the 16-Q

AM

signal set. Call this Fig-3.

(f) Repeat part (d) for the 16-Q

AM

signal set. Chose the rv to sym

bol mapping appropriately and explain how

you did it. Plot this result back in Fig-3.

(g) With G

ray-coding done, compare the sim

ulated bit error rate (BE

R) of Q

PS

K and 16-Q

AM

. (Hint: w

hile you are measuring the S

ER

in parts (e) and (f), you can also com

pute the BE

R for both the m

odulations). Plot these tw

o results together in Fig-4, again for Eb=

1 in both cases. Note that for com

puting BE

R, you w

ill

Instructions:S

ubmit only a single P

DF file for the report. N

ame the file as “rollnum

ber-assignment1-report.pdf”.

Your w

orking code, properly comm

ented, mu

st be inclu

ded

as Annexure-1 to the P

DF file, and also separately

sent by email to the T

As. Y

our working code can be nam

ed “rollnumber-assignm

ent1-code.m”. P

lease see other instructions, if any, in the W

hatsApp group.