Optimizing SNR with Matched Filter and QPSK Modulation

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QUESTION 1
Section A
A matched filter is an optimal linear filter used to maximize the SNR in the presence of
additional zero mean white Gaussian noise with power spectral density. Some of the common
applications are in radar where signals are sent out and the reflected signals are measured and
compared to the sent signal. They are also used in image processing as an improved SNR shows
an improved SNR for x-ray pictures.
The signal shows a rectangular pulse given as,
s ( t )= {A , for 0 ≤ t ≤T
0 , otherwise
The power spectral density is given as,
N 0
2 watts /Hz
Part 1
The matched filter output is y(t), and the filter input is given as, x(t),
x ( t ) =s ( t ) +n ( t ) 0 ≤ t ≤ T
The output is expressed as,
y ( t ) =s0 ( t ) +n (t)
2
S(t)
n(t)

The output has a combination of the signal and noise components of the input signal. To
determine the maximum peak pulse SNR,
η= |s0 ( t )|2
E [ n2 ( t ) ]
Part 2
The white noise is further described based on the spectral power density as,
H ( f )=2 K
N0
S¿ ( f ) e− j w t 0
When the input noise is the zero mean white Gaussian noise, the impulse response of the
matched filter is given as,
h(t)=Cs (t0−t )
h ( t )=F−1 [ H ( f ) ]= 2 K
N0
∫
−∞
∞
S¿ ( f ) e j w t0
e− j wt df
¿ 2 K
N0 [ ∫
−∞
∞
S¿ ( f ) e j 2 πf ( t0−t ) df ]
¿
¿ 2 K
N0
[ s ( t0 −t ) ]¿
Part 3
Since the signal S(t) is a real life signal, the maximum signal to noise ratio is given as,
2 K
N0
=Cs ...impulse response
The impulse response signal shows a slight backward shift of the original input signal before the
noise input.
Part 4
3

To show that
y ( T ) =∫
0
T
x ( τ ) h (T −τ )dτ
y ( T ) = [ x ( t ) , s ( t ) ] L2
Section B
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