# Luigi Paura » 7.Error probability for M-PSK

### Differential PSK

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”

### Differential PSK (next)

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.

### Differential PSK (next)

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.

### Error probability for differential B-PSK (D-BPSK)

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 rn (rn-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).

### Comparison between the memoryless modulation schemes (MMS)

The MMScan be compared in several ways. For example:
Which is the required ratio Eb/N0 to assure a given Pb for a fixed channel bandwidth 2W (or bit rate Rb) ?
Which is the minimum channel bandwidth ( or the maximum Rb ) to obtain a given Pb for a fixed Eb/N0

### Comparison between the memoryless modulation schemes (MMS) (next)

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.

### Comparison between the memoryless modulation schemes (MS) (next)

Efficient in power:
The PAM signals present a Pb which increases with M increasing (or k) for a fixed Eb/N0
To mantain Pb constant it is necessary to increase Eb/N0
The PAM signals are not efficient in power.
Analogously for M-PSK and QAM
The orthogonal signals exhibit a Pb which decreases with M (with k)
Analogously for simplex signals as well as biorthogonal ones

### Coded Transmissions

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

### Comparison between the memoryless modulation schemes (MS)

How do we pay the bandwidth and/or power saving?
We pay with the complexity increasing in transmission (coder) and in reception (decoder)

### Comparison between the memoryless modulation schemes (MS) (next)

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

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