563489578934

Problems 565
where u c is the start-up carrier phase, m(t) is the polar binary data baseband modulation, and
c(t) is a polar baseband spreading waveform that usually consists of a pseudonoise (PN) code.
The PN code is a binary sequence that is N bits long. The “bits” are called chips, since they do
not contain data and since many chips are transmitted during the time that it takes to transmit
1 bit of the data [in m(t)]. The same N-bit code word is repeated over and over, but N is a large
number, so the chip sequence in c(t) looks like digital noise. The PN sequence may be generated
by using a clocked r-stage shift register having feedback so that N = 2 r - 1. The autocorrelation
of a long sequence is approximately
R c (t) =a t T c
b
where T c is the duration of one chip (the time it takes to send one chip of the PN code). T c T b ,
where T b is the duration of a data bit.
(a) Find the PSD for the SS signal s(t). [Hint: Assume that m(t), c(t), and u c are independent.
In addition, note that the PSD of m(t) can be approximated by a delta function, since the spectral
width of m(t) is very small when compared to that for the spreading waveform c(t).]
(b) Draw a block diagram for an optimum coherent receiver. Note that c(t) m(t) is first coherently
detected and then the data, m(t), are recovered by using a correlation processor.
(c) Find the expression for P e .
★ 7–36 Examine the performance of an AM communication system where the receiver uses a product
detector. For the case of a sine-wave modulating signal, plot the ratio of [(S/N) out /(S/N) in ] as a
function of the percent modulation.
★ 7–37 An AM transmitter is modulated 40% by a sine-wave audio test tone. This AM signal is transmitted
over an additive white Gaussian noise channel. Evaluate the noise performance of this system
and determine by how many decibels this system is inferior to a DSB-SC system.
7–38 A phasing-type receiver for SSB signals is shown in Fig. P7–38. (This was the topic of
Prob. 5–21.)
(a) Show that this receiver is or is not a linear system.
(b) Derive the equation for SNR out of this receiver when the input is an SSB signal plus white
noise with a PSD of N 0 /2.
B
Low-pass filter
1,|f| < B
H(f) =
0
0, f elsewhere
C
Bandpass
input
r(t) A
D
–90°
phase shift
A 0 cos[ c t+¨c ]
Local
oscillator
+ Output
m ~ (t)
±
I
E
F
H(f)=
Low-pass filter
1,|f| < B 0
0, f elsewhere
G
Hilbert transform
–j, f > 0
H 2 (f) =
j, f < 0
H
Figure P7–38