Real time DSP

Professors: Eng. Diego Barral Eng. Mariano Llamedo Soria Julian Bruno

Filters

conventional 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!

Random Processes

random != deterministic concepts

realization ensemble ergodic

tools mean variance correlation/autocorrelation

stationary processes & WSS

Adaptive Filters

parts digital filter adaptive algorithm

filter FIR IIR (stability problems are difficult to handle)

Adaptive Filters

d(n) desired signal y(n) output of the filter x(n) input signal e(n) error signal

FIR Filter

wl(n) adaptive filter coefficients

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)

Performance Function

MSE surface One global minimum

point!

Gradient Based Algorithms

properties convergence speed steady-state performance computation complexity

method of steepest descent greatest rate of decrease (negative gradient) iterative (recursive)

LMS Algorithm

statistics of d(n) and x(n) are unknown estimation of MSE

avoids explicit computation of matrix inversion, squaring, averaging or differentiating

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

Performance Analysis

convergence speed large μ => fast convergence

λ => relation between stability and speed of convergence estimation

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

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

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

Applications

operate in an unknown enviroment track time variations

identification inverse modeling prediction interference canceling

Applications

adaptive system identification experimental modeling of a process or a plant

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

Applications

adaptive linear prediction

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)

Applications

adaptive noise cancellation (ANC)

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

Applications

adaptive channel equalization

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