Speech Enhancement - Digital Speech: Coding for Low Bit Rate Communication Systems - page 408

Assuming the system is linear and the echo path impulse response is of finite
length N , then the echo canceller forms the replica of the returned echo using,

N − 1

r(i) = ˆ a k y(i − k) (11.54)

k = 0

When a k 1 the returned and estimated echoes are

identical resulting in no residual echo. The coefficients of the transversal filter
are updated to match the slowly time-varying echo path impulse response
by minimizing the mean squared residual error given by:

= h k ,for k = 0 , 1 , ... ,N −

e 2 (i) r(i) ] 2

= − ˆ

[ r(i) (11.55)

When there is no near-end speech ( x(i) 0), the filter coefficients are updated

in such a way that the residual error tends to a minimum. The update of the
coefficients at each iteration is controlled by a step size β ,

=

h k (i + 1 ) = h k (i) + 2 βe(i)y(i − k) (11.56)

The convergence of the algorithm is determined by the step size β and the
power of the far-end signal y(i) . In general, making β large speeds up the
convergence, while a smaller β reduces the asymptotic cancellation error. It
has been shown that the convergence time constant is inversely proportional
to the power of y(i) and that the algorithm will converge very slowly for
low-signal levels [23]. To overcome this situation, the loop gain is usually
normalized by an estimate of the far-end signal power,

β 1
P y (i)

= =

2 β 2 β(i) (11.57)

where β 1 is a compromise value of the step size constant and P y (i) is an
estimate of the average power in y(i) at time i . The far-end signal power can
be estimated by

[ L y (i) ] 2 P y (i) = (11.58)

where,

L y (i + 1 ) = ( 1 − ρ)L y (i) + ρ | y(i) | (11.59)

2 − 7 . The above equation is only an estimate
of the average signal level, which is updated for every sample using the
approximation for ease of implementation in real-time.

and a typical value of ρ =