1. Chapter 6 Matched Filters
Matched filters for white noise
Integrate and Dump matched filter
Correlation processing
Huseyin Bilgekul
EEE 461 Communication Systems II
Department of Electrical and Electronic Engineering
Eastern Mediterranean University

2. Matched Filter The Matched Filter is the linear filter that maximizes:
Recall
Matched Filter
h(t)
H(f)
r(t)=s(t)+n(t)
R(f)
ro(t)=so(t)+no(t)
Ro(f)

3. Matched Filter
Design a linear filter to minimize the effect of noise while maximizing the signal.
s(t) is the input signal and s0(t) is the output signal.
The signal is assumed to be known and absolutely time limited and zero otherwise.
The PSD, Pn(f) of the additive input noise is also assumed to be known.
Design the filter such that instantaneous output signal power is maximized at a sampling instant t0, compared with the average output noise power:

4. Matched Filter The goal is maximize (S/N)out h(t) Threshold H(f)
Sampler
t = to
r(t)=s(t)+n(t)
R(f)
h(t)
H(f)
ro(t)=so(t)+no(t)
Ro(f)
Threshold
Detector
ro(t)=so(t)+no(t)
r(t)=s(t)+n(t)
so(t)
s(t) T T

5. Matched Filter
The matched filter does not preserve the input signal shape.
The objective is to maximize the output signal-to-noise ratio.
The matched filter is the linear filter that maximizes (S/N)out and has a transfer function given by:
where S(f) = F[s(t)] of duration T sec.
t0 is the sampling time
K is an arbitrary, real, nonzero constant.
The filter may not be realizable.

6. Signal and Noise Calculation
Signal output:
Output noise power or variance
Putting the pieces together gives:
Simplify Using Schwartz’ Inequality.
Equality occurs only if A(f) = K B*(f)

7. Signal and Noise Calculation
Apply the Schwartz Inequality:
Then we obtain:
Maximum (S/N)out is attained when equality occurs if we choose:
If p(t) is real, then we have conjugate symmetry P(-w)=P*(w)

8. Matched Filter for White Noise
For a white noise channel, Pn(f ) = No/2
Here Es is the energy of the input signal. The filter H(f ) is:
The output SNR depends on the signal energy Es and not on the particular shape that is used.
Impulse response is the known signal wave shape played “Backwards” and shifted by to.

9. Matched Filter for White Noise
Increase in the time-bandwidth product does not change the output SNR.
If a symbol lasts for T seconds, then there are 3 cases: (to< T, to= T and to> T)
to< T gives a NONCAUSAL input response
to> T gives a DELAY in deciding what was sent
to= T gives the MINIMUM DELAY for a decision plus it is REALIZABLE.
This is p(-t) delayed by t_m.

10. Impulse Response of Matched Filter
Thus, s(t) and h(t) have duration T.
The delay is also T
The output has duration 2T because s0(t) = s(t)*h(t).
Note that the peak value is at T.
This is p(-t) delayed by t_m.
so(t)
s(t)+n(t) 2T

11. Impulse Response of Matched Filter
The output is obtained by performing convolution s0(t) = s(t)*h(t).

12. MF Example for White Noise
Consider the set of signals:
Draw the matched filter for each signal and sketch the filter responses to each input
s1(t)
s2(t)
T/2 T
T/2 T

13. MF Example for White Noise
s2(t)
s1(t) T
T/2 T
T/2
h1(t)
h2(t)
T/2 T
T/2 T
T/2 T
y21(t)=s2(t)*h1(t)
T/2 T
y11(t)=s1(t)*h1(t)

14. Integrate and Dump (Matched) Filter

15. Integrate and Dump (Matched) Filter
Input Signal
Backward Signal
Matched Filter Impulse Response
Matched Filter Output Signal

16. Integrate and Dump Realization of Matched Filter

17. Correlation Processing

18. Correlation Processing
Theorem: For the case of white noise, the matched filter can be realized by correlating the input with s(t) where r(t) is the received signal and s(t) is the known signal wave shape.
Correlation is often used as a matched filter for Band pass signals.

19. Correlation (Matched Filter) Detection of BPSK