1. Professors: Eng. Diego Barral Eng. Mariano Llamedo Soria Julian Bruno
Real time DSP
Professors:
Eng. Diego Barral
Eng. Mariano Llamedo Soria
Julian Bruno

2. Filters conventional filters adaptive filters time-invariant
fixed coefficients
adaptive filters
time varying
variable coefficients
adaptive algorithm
function of incoming signal
exact filtering operation is unknown or is non-stationary!

3. Random Processes random != deterministic concepts tools
realization
ensemble
ergodic
tools
mean
variance
correlation/autocorrelation
stationary processes & WSS

4. Adaptive Filters parts filter digital filter adaptive algorithm FIR
IIR (stability problems are difficult to handle)

5. Adaptive Filters d(n) desired signal y(n) output of the filter
x(n) input signal
e(n) error signal

6. FIR Filter
wl(n) adaptive filter coefficients eq

7. Performance Function
coefficients are updated to optimize some predetermined performance criterion
mean-square error (MSE)
for FIR
R: input autocorrelation matrix
p: crosscorrelation between d(n) and x(n) eq eq

8. Performance Function
MSE surface
One global minimum point!
Fig 8.4

9. Gradient Based Algorithms
properties
convergence speed
steady-state performance
computation complexity
method of steepest descent
greatest rate of decrease (negative gradient)
iterative (recursive) eq

10. LMS Algorithm statistics of d(n) and x(n) are unknown
estimation of MSE
avoids explicit computation of matrix inversion, squaring, averaging or differentiating Eq Eq Eq Eq

11. Performance Analysis stability constraint
μ controls the size of the incremental correction
λmax is the largest eigenvalue of the autocorrelation matrix R
Px input signal power
large filters => small μ
strong signals => small μ
Eq 8.3.1
Eq 8.3.5

12. Performance Analysis convergence speed
large μ => fast convergence
λ => relation between stability and speed of convergence
estimation
Eq 8.3.6
Eq 8.3.8

13. Performance Analysis excess mean-square error
the gradient estimation prevents w from staying at wo in steady state
w varies randomly about wo
trade-off between the excess MSE and the speed of convergence
trade-off between real-time tracking and steady-state performance
Eq 8.3.9

14. Modified LMS Algorithms
normalized LMS algorithm
μ varies with input signal power
optimize the speed of convergence and maintain steady-state performance
independent of reference signal power
c is a small constant
μ(n) is bounded
0 < α < 2
eq 8.4.4

15. Modified LMS Algorithms
leaky LMS algorithm
insufficient spectral excitation may result in divergence of the weights and long term instability
where v is the leakage factor
0 < v ≤ 1
equivalent of adding low-level white noise
degradetion in performance
(1 - v) < μ
eq 8.4.5

16. Applications operate in an unknown enviroment track time variations
identification
inverse modeling
prediction
interference canceling

17. Applications adaptive system identification
experimental modeling of a process or a plant
fig 8.6

18. Applications adaptive linear prediction
provides an estimate of the value of an input process at a future time
in y(n) appear the highly correlated components of x(n)
i. e. speech coding and separating signals from noise
output is e(n) for spread spectrum corrupted by an additive narrowband interference

19. Applications
adaptive linear prediction
fig 8.7

20. Applications adaptive noise cancellation (ANC)
most signal processing techniques are developed under noise-free assumptions
the reference sensor is placed close to the noise source to sense only the noise, because noise from primary sensor and reference sensor must be correlated
the reference sensor can be placed far from the primary sensor to reduce crosstalk, but it requires a large-order filter
P(z) represents the transfer function between the noise source and the primary sensor
uses x(n) to estimate x’(n)

21. Applications
adaptive noise cancellation (ANC)

22. Applications adaptive channel equalization
transmission of data is limited by distortion in the transmission channel
channel transfer function C(z)
design of an equalizer in the receiver that counteracts the channel distortion
training of an equalizer
agreed sequence by the transmitter and the receiver
Decision device

23. Applications
adaptive channel equalization

24. Implementation considerations
finite-precision effects
prevent overflow
scaling of coefficients (or signal)
quantization & roundoff
=> excess MSE
=> stalling of convergence
depends on μ
threshold of e(n) -> LSB