Ch.2: 1, 4, 5, 9, 14, 17, 20, 22, 34

2.1 (1/2p )(Area) =(100x10^-6)² Þ A =0.5x10^-15

2.4 R(t ) = (1/2p )[1 + cos(w _ot ) + 4(t ² + 4)]

2.5 R(¥ )=0 Þ E[X(t)]=0 for all t. So, treating X as a scalar r.v.

then R(0)=4. So X~N(0,4) and Pr[X>4] = 0.0455.

2.9 X(t) =2sin(w t) where w ~ Uniform(a,b).

(a) E[X(t)]=(2/t)[cos(b)-cos(a)]=m (t)¹ constant for a=2, b=6.

So X(t) is not stationary.

(b) From (a) it follows that it is not ergodic.

(c) It is deterministic because given a value for w we know X(t) for all t.

2.14 (a) From Appendix A.2 S(w )=b s ² [ 1/{(w + w o)² + b ²} + 1/{(w -w o)² + b ²}]

2.17 (a) R_y(t )=E[(aX(t)+b)(aX(t+t )+b)]=a²Rx(t )+b²

(b) Similarly, R_xy(t )=aR_x(t )

2.20 Following (2.17) and noting that A and B are independent gives

Rxy(t )=Cov(A,B)cos(w t )/2

2.22 R_z(t )=R_x(t )R_y(t ) Þ S_z(w )=(1/2p )S_xÄ S_y(w )

where Ä is the convolution operation.

2.34

%Part (a): The program for generating the 256-sample realization is

%similar to that used in Problem 2.33* except for delta t.

%The rand statement used here is from MATLAB Version 3.5, and it

%results in a warning statement when using Version 4.0. This does

%not interfere with the solution. If you are using Version 4.0,

%the warning statement can be eliminated by deleting the

%rand('normal') statement in line 24 of the code and replacing

%rand with randn everywhere.

varwk=1-exp(-2*1);

sigmawk=sqrt(varwk);

phi=exp(-1);

%Now set up 256-point time sample as a vector called gtime.

gtime=zeros(1,256);

%Next generate the 256-point sample realization.

rand('normal')

gtime(1)=rand;

for i=1:255

gtime(i+1)=phi*gtime(i)+sigmawk*rand;

end

%Preview the sample realization for reasonableness.

plot(gtime)

title('Press ENTER to Continue')

pause

%Parts (b) and (c): We want the periodgrams for the 64-point, 128-

%point, and 256-point samples of gtime. We will first get the dft

%for these time samples and then form the periodograms from the dfts.

%dumxx's will be used as dummy variables. (Note that the scaling

%here will not be the same as in the text.)

dum64=gtime(1:64);

dft64=fft(dum64);

abdft64=abs(dft64);

pgram64=abdft64.^2;

%Now plot the periodogram (first term is for zero frequency).

k=0:1:63;

plot(k,pgram64,'o')

title('Press ENTER to Continue')

pause

%Now use similar code and plot the 128-point periodogram.

dum128=gtime(1:128);

dft128=fft(dum128);

abdft128=abs(dft128);

pgram128=abdft128.^2;

kk=0:1:127;

plot(kk,pgram128,'o')

title('Press ENTER to Continue')

pause

%Now use similar code and plot the 256_point periodogram.

dft256=fft(gtime);

abdft256=abs(dft256);

pgram256=abdft256.^2;

kkk=0:1:255;

plot(kkk,pgram256,'o')

title('Press ENTER to Continue')

pause

%Part (d): To do this part, begin with the periodogram for the

%256-point case. The variable pgram256 is a 1 x 256 row vector,

%and we need to average successive blocks of 8 elements of this

%vector and then plot the results. The plot will be truncated

%at the fold-over frequency.

avpgram=zeros(1,16);

for i=1:16

avpgram(i)=(sum(pgram256((8*i-7):(8*i))))/8;

end

%For comparison purposes on the plot, the spacing between samples

%should be 8 times that of the 256-point plot. Therefore, set kkkk

%accordingly with the appropriate offset from zero, and truncate

%the right half of the plot.

kkkk=4:8:252;

truncpgm=[avpgram zeros(1,16)];

plot(kkkk,truncpgm,'o')

title('Press ENTER to end Problem 2.34')