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## Impulse Response Measurement and Equalization

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**Impulse Response Measurement and Equalization**Digital SignalProcessing LPP Erasmus Program Aveiro 2012**Impulse response measurement**Measurement of impulse responses is a common task in different areas (specially important in audio processing) Typically, PC hardware is used to perform this task using the following setup: x(t) x[n] DAC h(t) y[n] ADC PC y(t) The following assumptions are made: The whole system between x[n] and y[n] shows time-invariant behaviour. All involved components are sufficiently linear The impulse response can be considered finite (h[n]≈0 for n > M)**Impulse response measurement**In theory, any input signal can be used to measure the impulse response of a linear system. The output signal can be expressed in terms of this impulse response by: Assuming that the impulse response only has N significant samples, the former equation can be rewritten in matrix form as: where L is the length of the input signal**Impulse response measurement**Then, the impulse response can be theoretically obtained as: where X-1* represents the pseudo-inverse of matrix X: (XT·X)-1·XTor XT·(X·XT)-1 However, in practice, very long input signals would be needed to attain high suppression of measurement noise, resulting in extremely large problems which may turn out to become computationally intractable.**Impulse response measurement**An atractive approach consist in using as input signal one whose auto-correlation function is similar to a delta sequence: In this case, the impulse response of the system can be obtained by cross-correlating the output and the input signals:**Impulse response measurement**Assuming that the length of the input signal is K and that the impulse response has only N significant samples, we have:**Impulse response measurement**Using this indirect approach, a great inmunity against noise is achieved: • The energy of the input signal can be as large as desired, since we have just to extend its duration. • As noise corrupting the output signal is uncorrelated with the input signal x, its contribution in the measurement of the impulse response is very small:**Impulse response measurement**Typical test signals: x(t) Rxx(t) Chirp (Continuous-time frequency varying sinusoid) PR sequence (13-bit Barker Code) White Gaussian Noise**Impulse response equalization**An Equalizer is a system whose main objective is to undo the effect of a previous stage, typically a transmission channel: Channel c[n] Equalizer g[n] + Ideally, the composite impulse response of the channel and the equalizer is a delta (flat frequency response), so that the output of the equalizer is a delayed version of the input signal. We can distinguish between two types of equalizers: Fixed equalizers (its coefficients do not change with time) Adaptive equalizers (its coefficients are updated periodically based on the current channel characteristics)**Impulse response equalization**Fixed Equalizer (Zero-Forcing) In zero-forcing equalization, the equalizer response g[n] attempts to completely inverse the channel by forcing : Or using matrix representation:**Impulse response equalization**Adaptive Equalizer When the channel is time-varying (LTV), it is necessary to update the equalizer coefficients in order to track the channel changes. Define the input signal to the equalizer as a vector ck where: and an equalizer weight vector gk, where Then, the output signal is given by: And the error signal ek is given by:**Impulse response equalization**Adaptive Equalizer An adaptive channel equalizer has the following structure.**Impulse response equalization**Adaptive Equalizer The least mean-square (LMS) algorithm carries out the mean squared error by recursively updating the coefficients using the following rules: The step parameter a, controls the adaptation rate and must be chosen carefully to guarantee convergence. The equalizer is converged if the error ek becomes steady.