The channel has introduced a linear distortion :
temporal dispersion since Th > T→ Intersymbol Interference (ISI) → frequency selectivity Hi(f) = Si(f) C(f)
Let us note that, since the signalis not band limited, only part of the transmitted energy can be received.
The maximum value of Eh, and, hence of the SNR, will be reached when W→∞; therefore
The best performances are obtained by assuring that the PSD of the transmitted signal be matched with the bandwidth of the channel C(f).
How do the performances change when we consider the transmission of a sequence of symbols instead of a single symbol?
The summation for n≠m represent the ISI.
The ISI effect can be visualized by means of oscilloscope, obtaining the so-called “eye pattern“. The closed eye denotes a high ISI level.
By adequately choosing both the transmitting filter GT(f) and the reception one GR(f) it is possible to satisfy the condition:
xn=0 per n≠0
thus nulling so the ISI. In this way, only the noise component which, however, depends on GR(f) will degrade the system performance.
There are necessary and sufficient conditions to remove ISI referred to as Nyquist conditions.
The presented analysis for baseband signals can be easily generalized to the transmission of bandpass signals, obtaining for them an expression which is equivalent for ISI.
If GT(f) exhibits nulls in f = m/T (example if gT(t) = rect(t/T).
There are not spectral lines!!→ Problems for the synchronization.
To make the PSD SV(f) compatible with the spectral channel requirements, we can modify:
X•c = q
Linear System in 2N+1 unknows (c-N, … cN) e 2N+1 with 2N+1 equations (see ex. pag. 544 Proakis-Salehi).
Let us emphasize that with an egualizer FIR it is not possible to equalize perfectly the channel in general, namely to remove the ISI completely but if N is increasing, ISI can be reduced arbitrarly.
τ = T/2 equalizer with a fractioned spacing
In such a case the samples at the receiving filter output (matched filter) can be acquired with a period sampling T/2.
Namely a double frequency with respect to the symbol spacing by halving the sampling interval for the fixed number of samples.
We can equalize in a double bandwidth namely (-1/T, 1/T).
By sampling the output of the receving filter with a double frequency we can avoid the aliasing.
We have to solve a linear system of 2N+1 equation with both the zero-forcing and MMSE criterion.
B•c = d
There are iterative procedures to avoid the inversion of the matrix B → stochastic gradient algorithm also referred to as Least Mean Square (LMS).
13. Cyclic Codes