Columbia University



RANDOM SIGNALS AND NOISE

ELEN E4815y

COLUMBIA UNIVERSITY

SPRING SEMESTER 2008

Problem Set # 8

Due Date: 17 April 2008

Problem #1

In this example we transmit an SSB-USB signal, but in this case the carrier at the receiver is not locked on to the phase of the input signal, i.e., there is a phase offset of ( radians (see figure on next page) between the receiver and the transmitter.

xSSB-USB(t) =s(t) cos (2(f0t) - s(t) sin (2(f0t)

a. Find the power spectral density of the receiver output signal, v(t). The result of this answer shows that even if there is an offset of ( radians, the output spectral density does not change.

b. Find the Signal-to-Noise ratio at the output, (S/N, out), of the SSB-USB receiver, and show that it is still equal to S/N, c=PT/N0W, i.e.,

S/N, out= S/N,c=PT/N0W

where PT is the received signal power.

The fact that neither the power spectral density nor the output signal-to-noise ratio changes, even if the detection is not coherent, is used in the transmission of voice signals. The ear, more or less, reacts to power spectral density, and if that stays the same (even if the detection is non-coherent) then the conversation is still understandable.

SSB Receiver

A cos 2(f0t n(t), N0/2 watts/Hz 2 cos (2(f0t+()

v(t) + n0(t)

s(t) xSSB-USB(t)

PT

nin(t)

HBPF(f) HLPF(f)

1 1

f f

0 f0 f0 +W 0 W

Pay attention to the fact that there is an offset of ( radians, in the carrier multiplication, at the receiver.

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SSB-USB

Modulator

BPF

LPF

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