1. COLOURED NOISE REMOVAL AND
EQUALISING THE CHANNEL EFFECT
FROM A NOISY AUDIO SIGNAL
G. Anudeep Reddy (EC08484)
1 G. Madhuri (EC08485)

2. INTRODUCTION
 We want to transmit a song signal.
Song Signal Transmitter
Loud Speaker Receiver
2

3. REMOVAL OF NOISE
 The signal may be corrupted by Noise.
Song
Transmitter Noise
Signal
Receiver
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4. REMOVAL OF NOISE
 The signal may be corrupted by Noise.
Song
Transmitter Noise
Signal
Loud
Filter Receiver
Speaker
4

5. WHITE NOISE
 White noise is a signal (or process), having
equal power in any band of a given bandwidth
(power spectral density).
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6. COLORED NOISE
 Based on Spectral density (power distribution
in the frequency spectrum) we can distinguish
different types of noise.
 This classification by spectral density is given
"color" terminology, with different types
named after different colors.
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7. COLORED NOISE
 The color names for these different types of
sounds are derived from a loose analogy
between the spectrum of frequencies of sound
wave present in the sound and the equivalent
spectrum of light wave frequencies.
 That is, if the sound wave pattern of "blue
noise" were translated into light waves, the
resulting light would be blue, and so on.
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8. PINK NOISE
 Similar to White Noise except the power
density decreases 3 dB per octave as the frequency
increases. In technical terms the
density is inversely proportional to the frequency.
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9. BLUE NOISE
 Similar to White Noise except the power
density increases 3 dB per octave as the frequency
increases. In technical terms the
density is proportional to the frequency.
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10. GENERATION OF COLORED NOISE
 Colored noise can be generated by passing the
white noise through a shaping filter.
 The response of the colored noise can be
varied by adjusting the parameters of the
shaping filter.
White Shaping Colored
Noise Filter Noise
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11. REMOVAL OF CHANNEL EFFECT
 The signal may be corrupted by channel
transfer function also.
Song
Transmitter Channel Noise
Signal
Receiver
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12. REMOVAL OF CHANNEL EFFECT
 The signal may be corrupted by channel
transfer function also.
Song
Transmitter Channel Noise
Signal
Loud
Filter Equalizer Receiver
Speaker
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13. EQUALIZER
 The equalizer should have transfer function
which is inverse of channel.
1
H (z)
C (z)
 Where H(z) is the transfer function of equalizer
and C(z) is the transfer function of channel.
 But in most of the cases we do not know the
transfer function of the channel, so we will
adapt the equalizer transfer function using
Learning algorithm.
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14. COMPLETE BLOCK DIAGRAM
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15. CHANNEL AND CHANNEL EQUALIZER
o A finite impulse response (FIR) filter is a type of a
signal processing filter whose impulse response (or
response to any finite length input) is of finite duration,
because it settles to zero in finite time.
o This is in contrast to infinite impulse response (IIR)
filters, which have internal feedback and may continue
to respond indefinitely (usually decaying).
o The impulse response of an Nth-order discrete-time
FIR filter, lasts for N+1 samples, and then dies to zero.
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16. o FIR filters can be discrete
time or continuous-time, and digital or analog.
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17.  For a discrete-time FIR filter, the output is a
weighted sum of the current and a finite number of
previous values of the input. The operation is
described by the following equation, which defines the
output sequence y[n] in terms of its input
sequence x[n]:
o The channel can be considered as a discrete
time digital FIR filter
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18. o Similarly the equalizer can be considered as an FIR
filter(discrete time, digital FIR filter)
 From the block diagram, it is evident that the optimal
equalizer should have transfer function which is inverse
of channel. Hence channel equalization is also known as
inverse filtering.
Transfer function of Channel , C(z)= b0+ b1z-1+ b2z-2 + bnz-n
Transfer function of the Equalizer, H(z)= 1/C(z)
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19. AN ADAPTIVE LINEAR EQUALIZER
xk-1 xk-2 xk-L+1
xk Z-1 Z-1 Z-1
w0k w1k w2k w(L-1)k
∑
yk
There is an input signal vector, x 0 , x1 ... x L 1
a corresponding set of adjustable weights, w 0 , w1 ... w L 1
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a summing unit, and a single out put signal.

20. ADAPTIVE LINEAR EQUALIZER
o A procedure for adjusting or adopting the weights is
called weight adjustment or adaptation procedure.
o The combiner is called linear because for fix setting of
weights its output is a linear combination of the input
components.
o The output of the combiner can be represented as
L
yk w lk x k l
l 0
where w lk denotes l th weight at k th instant.
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21. If the weight and input vectors are expressed as
T
Xk [ x0 k x1 k  x ( L 1) K
]
T
Wk [ w0 k w1 k  w ( L 1) K
]
then the output is given by
T
yk xk wk
The weights of the combiner are to be updated
using various learning algorithms
Ravi Kumar Jatoth Department 21
of ECE NITW

22. LEARNING ALGORITHMS
 LMS Algorithm
 RLS Algorithm
 Kalman Filter
 Neural Algorithm
 Fuzzy Logic System
 Optimization Algorithms
All the algorithms update the weights of the
equalizer using different cost functions. 22
Ravi Kumar Jatoth Department
of ECE NITW

23. LEAST MEAN SQUARE ALGORITHM
❏ LMS: adaptive filtering algorithm having
two basic processes
✔ Filtering process, producing
1) output signal
2) estimation error
✔ Adaptive process, i.e., automatic
adjustment of filter tap weights
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24. LEAST MEAN SQUARE ALGORITHM
o LMS algorithm is one of the conventional
techniques applied to channel equalization. The cost
function is Mean Square Error (MSE). It updates the
weights of the adaptive FIR filter based on the error
obtained. The instantaneous error at any time-step 'k'
can be represented as
e(k) = d(k) – y(k)
where d(k) delayed input reference is signal at time-
step „k‟, and 'y(k)‟ is estimated output from equalizer.
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25. o The equalizer filter's impulse response vector is
adapted using the following equation,
w(k+1) = w(k) + 2µ.e(k).x(k)
where µ is called „Convergence factor’ or ‘Learning
rate parameter’, (0 ≤ µ ≤ 1).
x(k) Is input from transmitter at time-step 'k'.
o This procedure is repeated till the Mean Square Error
(MSE) of the network approaches a minimum value.
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26. STABILITY OF LMS
 More practical test for stability is
2
0
input signal power
 Larger values for step size
 Increases adaptation rate (faster adaptation)
 Increases residual mean-squared error
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351M Digital Signal Processing

27. xk-1 xk-2 xk-L+1
xk Z-1 Z-1 Z-1
w0k w1k w2k w(L-1)k
∑
ek yk
LMS -
Algorithm ∑
+
dk
Fig. 2 Adaptive filter using LMS algorithm
T
Xk xk xk 1
 xk L 1 the L-by-1 tap input vector.
T
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Wk w0 k w1 k  w L 1 k the L-by-1 tap weight vector

28. LMS BLOCK DIAGRAM
a0
x(k)
Z-1 a1
Z-1
a2 y (k) +
∑ ∑ e (k)
-
Z-1 a3 d(k)
Z-1 a4
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LMS

29. NUMERICAL EXAMPLE- CHANNEL EQUALIZATION
❏ Transmitted signal: random sequence of
±1‟s.
❏ The transmitted signal is corrupted by a
channel.
❏ Channel impulse response:
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30. ❏ The amplitude distortion, and eigen value spread,
were controlled by W.
 The received signal is processed by a linear, 11-tap
FIR equalizer adapted with the LMS algorithm
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31. 31

32. REFERENCES
 “Digital Signal Processing using MATLAB” demos
by Charulatha Devi.
 Georgi Illiev and Nikola Kasabov, "Channel
Equalization using Adaptive Filtering with
Averaging", University of Otago, Newzeland.
 M Reuter, J Zedlier, "Nonlinear effects in LMS
adaptive equalizers", IEEE Trans.Signal
Processing, June1999.
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