Luigi Paura » 9.Transmission through bandlimited AWGN channels

Transmission through bandlimited AWGN channels

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

Model of bandlimited channel

Model of bandlimited channel

Model of bandlimited channel

Model of bandlimited channel

Model of bandlimited channel

Model of bandlimited channel

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)

Example

Example

Example

Example

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?

Transmission of PAM signals through baseband channels with band limitation

Transmission of PAM signals through baseband channels with band limitation

Transmission of PAM signals through baseband channels with band limitation

Intersymbol Interference (ISI)

Intersymbol Interference (ISI)

By adequately choosing both the transmitting filter G_T(f) and the reception one G_R(f) it is possible to satisfy the condition:

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.

Intersymbol Interference (ISI)

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.

A recall on spectrum evaluation for digital modulated signals

A recall on spectrum evaluation for digital modulated signals

A recall on spectrum evaluation for digital modulated signals

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)

Signal design for bandlimited channel  Nyquist Criterion

Signal design for bandlimited channel  Nyquist Criterion

Signal design for bandlimited channel  Nyquist Criterion

Signal design for bandlimited channel  Nyquist Criterion

Signal design for bandlimited channel  Nyquist Criterion

Signal design for bandlimited channel  Nyquist Criterion

Signal design for bandlimited channel  Nyquist Criterion

PAM signal transmission through baseband channels with bandwidth limitation

PAM signal transmission through baseband channels with bandwidth limitation

PAM signal transmission through baseband channels with bandwidth limitation

Evaluation of P(e)

Equalization

Equalization

Equalization

Equalization

Equalization

Equalization

Equalization

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.

Equalization

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.

Equalization

Equalization

Equalization

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).