Error signals inverse filtering

Here we have expressed x n purely in terms of the coefficients and the current and previous values ofy[ ] if we compare this to equation 10.54, we see that this is just a standard FIR filter. Hence if we have the speech signal and the coefficients for a single frame, we can find x n by a standard FIR filter operation. Apart from a gain normahsation, this will be identical to the error signal x[ ]. The inverse filtering technique is usefiil because often we are required to find the error signal after the LP analysis has been performed, or for cases (explained below) where we wish to... 

The received signal can be passed through a matched filter, a delay line inversely related to frequency, an autocorrelater, or any technique to produce an equivalent pulse length equal to the reciprocal of the total bandwidth transmitted. The amplitude of this compressed signal is equal to the square root of the time-frequency-bandwidth product of the transmitted signal. The errors are limited to system component phase and frequency stabilities and linearities over the times to transmit and to receive. 

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