**Summary**:

- Channel Model
- Transmission of PAM signals through baseband channels
- Transmission of PAM signals through bandpass channels
- Brief outline on Spectra of digital modulated signals
- Design of signalling schemes for bandlimited channels
- Outline on P(e) evaluation
- Equalization

The channel has introduced a linear distortion :

temporal dispersion since T_{h} > T→ **Intersymbol Interference (ISI)** → frequency selectivity H_{i}(f) = S_{i}(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 *E _{h}*, and, hence of the SNR, will be reached when

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 *G _{T}*(

*x*_{n}=0 per *n*≠0

thus nulling so the ISI. In this way, only the noise component which, however, depends on G_{R}(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 G_{T}(f) exhibits nulls in f = m/T (example if g_{T}(t) = rect(t/T).

There are not spectral lines!!→ Problems for the synchronization.

To make the PSD S_{V}(f) compatible with the spectral channel requirements, we can modify:

- S
_{a}(f) – line coding (example: AMI Coding) - G
_{T}(f) spectrum of the carrier pulse g_{T}(t)

**X•c = q**

Linear System in 2N+1 unknows (c_{-N}, … c_{N}) 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**

c_{opt}= B^{-1}•d

There are iterative procedures to avoid the inversion of the matrix B → stochastic gradient algorithm also referred to as Least Mean Square (LMS).

*5*. Digital Transmission over AWGN chanel

*6*. Evaluation of P(e) for optimum RX in AWGN

*7*. Error probability for M-PSK

*8*. Noncoherent ML Demodulation of FSK signals in AWGN

*9*. Transmission through bandlimited AWGN channels

*13*. Cyclic Codes

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