EE513
Audio Signals and Systems
Wiener Inverse Filter
Kevin D. Donohue
Electrical and Computer Engineering
University of Kentucky
Weiner Filters
A class of filters, referred to as Wiener filters, exploit
correlation information between signal and noise to
enhance SNR or reduce distortion. The Wiener filter is the
optimal filter for enhancing SNR of a random signal in
random noise. Signals and noise are characterized by their
PSDs or ACs, and objective metrics are either SNR
enhancement or minimization of least-square error.
These filters are named after Norbert Weiner:
http://en.wikipedia.org/wiki/Norbert_Wiener
Wiener Filters and Noise
Let s[n] be the original signal and y[n] be the corrupted
version. The error signal or noise is given by:
[n]  y[n]  s[n]
Minimizing the error in the L2 or mean square error (MSE)
sense means minimizing the expected value of:



E  [n]  E y[n]  s[n]
2
2

This is equivalent to maximizing the SNR:


E  [ n] 
2


E s[n]
2

E y[n]  s[n]
2

E s[n]
 

2

Signal Power

E  2 [ n]
Noise Power
Wiener Filter Objective
Let w[n] be the filter to maximize SNR or equivalently to minimize the
MSE:


~
y [ n] 
 y[m]w[n  m]   s[m]  [m]w[n  m]
m  
m  
Express MSE in terms of the above equation:


2
 

~
E y[n]  s[n   ]  E   s[m]  [m]w[n  m]  s[n   ] 
 m

where  is a delay parameter to relax a causality constraint and
2
typically improve performance. The first equation can be expressed
more directly in the frequency domain:


~ˆ
ˆ [k ] Wˆ [k ]
Y [k ]  Sˆ[k ]  
Wiener Filter and SNR
Assuming the signal and noise are uncorrelated, zero-mean stationary
processes, it can be shown that the optimal filter for minimizing MSE
is:
2

ˆ
E S[k ] 


Wˆ [k ] 
2
2



ˆ
ˆ
E [k ]  E S[k ] 




PSD of Noise
PSD of Signal
Can also be rewritten as:
Wˆ [k ] 
1
2

ˆ
E [k ] 


1
2


E Sˆ[k ]



1
1
1
SNR[ k ]
Homework 1
Download mat file (wienerhw1.mat) from:
http://www.engr.uky.edu/~donohue/ee513/data/wienerhw1.mat
It will contain the signal vectors described below with associated sampling frequency fs.
Examples of the signal and noise processes are stored in vectors sig and nos with
normalized power.
A) Plot the spectral magnitude of the Wiener filter for a signal plus noise process
assuming a signal-to-noise ratio of -15 dB, 0 dB, and 15 dB. Describe how the SNR
changes the spectral shape of the filter. Describe how this change makes sense for an
optimal filter for this type. Hand in commented code and a clearly labeled plot, and the
requested descriptions.
B) Apply a Wiener filter to the data in vector sigpnos, which is a combination of the
signal and noise from the same source as the examples. Note that you do not know the
SNR for this case. Since you know the PSD shapes you can try to assess the SNR by
examining the PSDs or ACs of combined signal. Also assume that the SNR is between 25 and 0 dB and create a loop to increment through various levels of SNR and listen to
or test the result to determine at what level the best performance. Hand in the
commented code used to filter and test the signal and determine the best SNR for the
filter design. Clearly indicate the SNR you thought was the best.
FIR Inverse (Wiener) Filters
An inverse filter undoes distortions due to frequency
selective channels/systems and restore the original
transmitted/driving signal. This type of filtering is
sometimes referred to as deconvolution. Let h(n) denote
the impulse response of the channel/system. The inverse
filter, hI(n), is described by:
h(n) * hI (n)   (n)
Hˆ ( z ) Hˆ ( z )  1
I
FIR Inverse Filters - Polynomial Division
Assume that for practical purposes the channel/system can
be modeled as an all-pole system, therefore the inverse
filter is an all-zero system. A direct way of obtaining the
impulse response of the inverse filter, hI(n), is to expand
rational polynomial Hˆ ( z ) through long division and
truncate the sequence after M+1 coefficients:

M

1
k
k
H I (z) 
  hI (k )z   bk z   bk z  k
H ( z ) k 0
k 0
k  M 1
The resulting error becomes:

E   bI2 (n)
2
n  M 1
FIR Inverse Filters – Least Squares
Another design can be obtained via a least-squares
approach:
h(n)
FIR Filter
{b k}
-
d ( n)   ( n)
e(n)
{ bk}
Minimize Sum of
Squared Errors
where d(n) is the desired response and the error of the filter
output is e(n). The error and overall squared error E2 are given
by:
2
M
e(n)  d (n)   bk h(n  k )
k 0
M


E   d (n)   bk h(n  k )
n 0 
k 1


2
FIR Inverse Weiner Filter
After minimizing E2 (take the derivatives with respect to
each bk and set the result to zero), it can be shown that the
optimal set of {bk}’s are the solution to the M equations
given by:
M
 bk rhh (k  l )  rdh (l )
k 1
l  0,1,2, M
where rhh(.) is the autocorrelation for h(n), and rdh(.) is the
cross-correlation between h(n) and d(n):

rhh (l )   h(n)h(n  l )
n0

rdh (l )   d (n)h(n  l )
n0
FIR Inverse Filters – Least Squares
For the special case where d(n) = (n):
 h(0)
rdh (l )  
0
for l = 0
otherwise
Therefore, the following system of equations can be used
to solve for the filter coefficients:
rhh (1)
 rhh ( M )  b0  h(0)
 rhh (0)
 r (1)
 b   0 
rhh (0)
hh

 1   


      
 
r ( M ) r ( M  1)  r (0)  b   0 

 hh
 M  
hh
hh
Matrix is Symmetric and Toeplitz, can use LevensonDurbin algorithm to solve
Example
Given desired response (original input to the system) d(n) and the
actual response of system h(n) up to length N, design an Mth order FIR
(Wiener) inverse filter. Create the following vector and matrix:
d   d (0) d (1)  d ( N )
T
0
 h( 0 )
 h(1)
h( 0 )



h( M  1)
H   h( M )

h( M  1) h( M  2)




 h( N )
h( N  1)










 h( N  M )


0
0
0
h( 0 )
h(1)
Then compute desired filter coefficients by solving the following matrix
equation for b:
H T Hb  H Td
where
b  b0
b1  bM 
T
Example
Then test for stability (was original system minimum phase?), apply
h(n) (plus a little noise, less than -3 dB) to the inverse filter. If the
result is anomalous (not close to the signal of interest), add a delay to
the desired response and repeat (i.e. use):
d  0 ....0 d (0) d (1)  d ( N  1)
T
Insert 0’s to delay desired response. This provides the filter with more
degrees of freedom to undo the system response at the expense of delaying
the output.
Increasing filter order can also improve performance.