Matched Filters

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```					Matched Filters

By: Andy Wang
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
   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.
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.
x (t )  g (t )  w(t ),         0t T
y (t )  g o (t )  n(t )
Signal              x(t)    LTI filter of impulse   y(t)            y(T)
∑                   response
g(t)
h(t)                 Sample at
Linear receiver              time t = T
White noise
w(t)
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:

2                2
| g o (T ) |         | g o (T ) |
SNR                         
n   2
  2
E n (t )      
where |go(T)|2 is the instantaneous power of the filtered signal, g(t) at
point t = T, and σn2 is the variance of the white gaussian zero mean
filtered noise.
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

Go ( f )  G ( f ) H ( f )
so g o (t )   G ( f ) H ( f )e j 2ft df
then | g o (t ) |2 |  G ( f ) H ( f )e j 2ft df |2
Deriving the matched filter (4/8)
   Examine σn2:

 
 n 2  E nt 2  E nt but this is zero mean so
  E nt  
n
2               2
and recall that

E nt    varnt   R 0  autocorrelation at   0
2
n

Rn           S n  f e j 2f df autocorrelation is inverse
Fourier transform of power
Rn 0          S  f  1 df
n
spectral density
Deriving the matched filter (5/8)
   Recall:
filter
SX(f)                   H(f)                SX(f)|H(f)|2 = SY(f)
No
 In this case, SX(f) is PSD of white gaussian noise, S X ( f )  2
   Since Sn(f) is our output:
| H  f  |2
No
Sn ( f ) 
2
 n 2  E n(t ) 2   Rn 0          | H  f  |2 df  o  | H  f  |2 df so
No                    N
2                     2
|  H ( f )G ( f )e j 2fT df |
SNR 
| H  f  |2 df
No
2 
Deriving the matched filter (6/8)
   To maximize, use Schwartz Inequality.

 | 1 x  |2 dx       Requirements: In this
case, they must be finite

 |  2 x  |2 dx  
signals.
                 
|  1 x  2 x dx |2   | 1 x  |2 dx  |  2 x  |2 dx
               
This equality holds if φ1(x) =k φ2*(x).
Deriving the matched filter (7/8)
   We pick φ1(x)=H(f) and φ2(x)=G(f)ej2πfT and want to make the
numerator of SNR to be large as possible
|  H ( f )G ( f )e 2fT df |  | H ( f ) |2 df   | G ( f )e j 2fT |2 df
|  H ( f )G ( f )e 2fT df |     | H ( f ) |2 df  | G ( f ) |2 df

No                            No
2                             2 
2
| H ( f ) | df                   | H ( f ) |2 df

SNR 
 | G ( f ) |2 df 2 | G ( f ) |2 df maximum SNR
                       according to Schwarz
No                No              inequality
2
Deriving the matched filter (8/8)
   Inverse transform
   Assume g(t) is real. This means g(t)=g*(t)
   If F g (t )  G  f 
F g * (t )  G *  f 
then G f   G *  f  for real signal g(t)
G *  f   G f  through duality
   Find h(t) (inverse transform of H(f))
h(t )  k  G  f e  j 2fT e j 2ft df

 k  G  f e  j 2f T t df      h(t) is the time-reversed and
delayed version of the input
 k  G  f e j 2f T t df         signal g(t).

ht   kgT  t                            It is “matched” to the input signal.
What is a correlation detector? (1/1)
   A practical realization of the   Detector
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,
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, m .ˆ        Signal Transmission Decoder
The equivalence of correlation and
   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:

y j t      x h t   d
j

The equivalence of correlation and
   From the definition of the
matched filter, we can
incorporate the impulse
hj(t) and the input signal   h j t    j T  t 
φj(t) so that:                            

   Then, the output             y j (t )     x  T  t   d

j
becomes:                                  
   Sampling at t = T, we get:   y j t      x   d
j

The equivalence of correlation and
Matched filters
   So we can see that the
detector part of the
implemented using either
matched filters or
correlators. The output of
each correlator is            Correlators
equivalent to the output of
a corresponding matched
filter when sampled at t =
T.

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