082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

24.15 Discrete convolution and correlation 817
Eitherdirectly,orusingacomputerpackage,takingtheinversed.f.t.yieldsthesequence
39,−129,78,48
which is the required circular correlation. You may like to verify this by directly calculating
f ○⋆ g.
Itfollowsfromthetheoremthatthecircularautocorrelationsequencehasthefollowing
property:
D{f ○⋆ f}=F[k]F[k]
Example24.34 Usethe d.f.t. tofind the circularautocorrelationofthe sequence f[n] = 8,3,−1,2.
Solution Itisstraightforward buttedioustoshow thatthe d.f.t.of f[n] is
F[k]=12,9−j,2,9+j
Then,the conjugate ofF[k] isF[k]=12,9+j,2,9−j,and
F[k]F[k] = 144,82,4,82
Finally,takingtheinversed.f.t.givesthesequence78,35,−4,35whichyoucanverify
isthe circular autocorrelation of f[n] = 8,3,−1,2.
Engineeringapplication24.5
Useofcorrelationinradar
Both sonar and radar operate by transmitting a signal which bounces off a distant
target and returns to a receiver. The main difference is that sonar uses sound waves
whereasradarusesradiowaves.Thetimetakenforthesignaltoreturntothereceiver
having reached the target can be used to deduce,d, the distance between the transmitter
and the target. In both radar and sonar, the received signal is typically very
much smaller than the transmitted signal and can contain a lot of unwanted noise.
High-gain electronic amplifiers may be required to make the signal large enough to
be processed and these introduce even more noise. In addition the receiver experiences
interference from other sources which has to be separated from the returning
signal. Correlation techniques are very useful for retrieving a signal from the noise
and the interference.
We will consider a radar system. The signals transmitted are usually modulated
usingahigh-frequencycarriersignal(seeEngineeringapplication24.1).Forourpurposesthe
presence ofacarrier signalisnotimportant.
However, the high-frequency component of the signal can be removed
electronically prior to the signals being sampled by a process called
demodulation.
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