If the phase is:

ø’=ø+π`→2`

ø’=2ø’+2π=2ø’mod2π`→`

therefore there is a π ambiguity in the phase estimate

Such ambiguity destroys the information which is carried by a phase of “0″ or “π”

If the information is carried by the phase difference associated with two adiacent symbols rather than the phase associated with a single symbol the ambiguity is removed.

By means of a differential coding the ambiguity problem has been solved.

Both the differential coding and decoding are implemented by simple logic circuits which precede the modulator and follow the detector, respectively.

If a coherent receiver is utilized, the P(e) is roughly double the P(e) obtained with the B-PSK (non differential).

This result is justified by considering that if a symbol interval is highly corrupted by noise, such an event can cause two errors in two consecutive decisions.

If a non-coherent receiver is utilized which performs the decisions by resorting to r_{n} (r_{n-1})*, it can be shown that the decision variable can be approximated with the one associated with the coherent receiver provided that the noise component exhibits a double power → **3dB loss in SNR (OK for M > 4)**.

The MMScan be compared in several ways. For example:

Which is the required ratio E_{b}/N_{0} to assure a given P_{b} for a fixed channel bandwidth 2W (or bit rate R_{b}) ?

Which is the minimum channel bandwidth ( or the maximum R_{b} ) to obtain a given P_{b} for a fixed E_{b}/N_{0}

How to evaluate the bandwidth of a modulation scheme?

The **Shannon bandwidth** gives a measure of the spectral occupacy without requiring the knowledge of the specific signal waveform.

Efficient in power:

The PAM signals present a P_{b} which increases with M increasing (or k) for a fixed E_{b}/N_{0}

To mantain P_{b} constant it is necessary to increase E_{b}/N_{0}→

The PAM signals are not efficient in power.

Analogously for M-PSK and QAM

The orthogonal signals exhibit a P_{b} which decreases with M (with k)

Analogously for simplex signals as well as biorthogonal ones

Do signaling schemes exist which are both efficient in power and in bandwidth?

**Yes**

How do we pay the bandwidth and/or power saving?

We pay with the complexity increasing in transmission (coder) and in reception (decoder)

The comparison must take into account also the complexity of the devices in transmission and in reception.

Example:

The PAM signals require a single correlator.

The orthogonal signals require (in general) M correlators

*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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