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Luigi Paura » 7.Error probability for M-PSK

Error probability for M-PSK

Error probability for M-PSK (next)

Demodulation for bandpass signals

Synchronization

Differential PSK

If the phase is:

ø’=ø+π→2ø’=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.

Non-coherent demodulation for differential PSK

Non-coherent demodulation for differential PSK (next)

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 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).

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

Error Probability for QAM

Comparison between QAM and M-PSK

Union Bound

Union Bound (next)

Union Bound

Comparison between the memoryless modulation schemes (MMS)

The MMScan be compared in several ways. For example:
Which is the required ratio E_b/N₀ 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/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 (MMS) (next)

Comparison between the memoryless modulation schemes (MS)

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

Efficient in power:
The PAM signals present a P_b which increases with M increasing (or k) for a fixed E_b/N₀
To mantain P_b constant it is necessary to increase E_b/N₀→
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

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

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