Matched%20Filters

1
Matched Filters

• By Andy Wang

2
What is a matched filter? (1/1)

• A matched filter is a filter used in
  communications to match a particular transit
  waveform.
• It passes all the signal frequency components
  while suppressing any frequency components where
  there is only noise and allows to pass the
  maximum amount of signal power.
• The purpose of the matched filter is to maximize
  the signal to noise ratio at the sampling point
  of a bit stream and to minimize the probability
  of undetected errors received from a signal.
• To achieve the maximum SNR, we want to allow
  through all the signal frequency components, but
  to emphasize more on signal frequency components
  that are large and so contribute more to
  improving the overall SNR.

3
Deriving the matched filter (1/8)

• A basic problem that often arises in the study of
  communication systems is that of detecting a
  pulse transmitted over a channel that is
  corrupted by channel noise (i.e. AWGN)
• Let us consider a received model, involving a
  linear time-invariant (LTI) filter of impulse
  response h(t).
• The filter input x(t) consists of a pulse signal
  g(t) corrupted by additive channel noise w(t) of
  zero mean and power spectral density No/2.
• The resulting output y(t) is composed of go(t)
  and n(t), the signal and noise components of the
  input x(t), respectively.

LTI filter of impulse response h(t)
Signal g(t)
y(t)
x(t)
y(T)
?
Sample at time t T
Linear receiver
White noise w(t)
4
Deriving the matched filter (2/8)

• Goal of the linear receiver
• To optimize the design of the filter so as to
  minimize the effects of noise at the filter
  output and improve the detection of the pulse
  signal.
• Signal to noise ratio is

where go(T)2 is the instantaneous power of the
filtered signal, g(t) at point t T, and sn2 is
the variance of the white gaussian zero mean
filtered noise.
5
Deriving the matched filter (3/8)

• We sampled at t T because that gives you the
  max power of the filtered signal.
• Examine go(t)
• Fourier transform

6
Deriving the matched filter (4/8)

• Examine sn2

but this is zero mean so and recall that
autocorrelation at
autocorrelation is inverse Fourier transform of
power spectral density
7
Deriving the matched filter (5/8)

• Recall

filter
H(f)
SX(f)
SX(f)H(f)2 SY(f)

• In this case, SX(f) is PSD of white gaussian
  noise,
• Since Sn(f) is our output

8
Deriving the matched filter (6/8)

• To maximize, use Schwartz Inequality.

Requirements In this case, they must be finite
signals.
This equality holds if f1(x) k f2(x).
9
Deriving the matched filter (7/8)

• We pick f1(x)H(f) and f2(x)G(f)ej2pfT and want
  to make the numerator of SNR to be large as
  possible

maximum SNR according to Schwarz inequality
10
Deriving the matched filter (8/8)

• Inverse transform
• Assume g(t) is real. This means g(t)g(t)
• If

then
for real signal g(t)
through duality

• Find h(t) (inverse transform of H(f))

h(t) is the time-reversed and delayed version of
the input signal g(t). It is matched to the
input signal.
11
What is a correlation detector? (1/1)

• A practical realization of the optimum receiver
  is the correlation detector.
• The detector part of the receiver consists of a
  bank of M product-integrators or correlators,
  with a set of orthonormal basis functions, that
  operates on the received signal x(t) to produce
  the observation vector x.
• The signal transmission decoder is modeled as a
  maximum-likelihood decoder that operates on the
  observation vector x to produce an estimate, .

Detector
Signal Transmission Decoder
12
The equivalence of correlation and matched filter
receivers (1/3)

• We can also use a corresponding set of matched
  filters to build the detector.
• To demonstrate the equivalence of a correlator
  and a matched filter, consider a LTI filter with
  impulse response hj(t).
• With the received signal x(t) used as the filter
  output, the resulting filter output, yj(t), is
  defined by the convolution integral

13
The equivalence of correlation and matched filter
receivers (2/3)

• From the definition of the matched filter, we can
  incorporate the impulse hj(t) and the input
  signal fj(t) so that
• Then, the output becomes
• Sampling at t T, we get

14
The equivalence of correlation and matched filter
receivers (3/3)
Matched filters

• So we can see that the detector part of the
  receiver may be implemented using either matched
  filters or correlators. The output of each
  correlator is equivalent to the output of a
  corresponding matched filter when sampled at t
  T.

Correlators