S Digital Communication Systems Bandpass modulation II

Timo O. Korhonen, HUT Communication Laboratory Matched filtering and correlator reception u Note that both circuits fulfil the expression Correlator (or integrate & dump receiver) Matched filter

Timo O. Korhonen, HUT Communication Laboratory

Coherent reception SNR and error rate u Express energy / bit embedded in signaling waveforms by u Therefore SNR for any coherent carrier modulation is Note that signaling waveform correlation greatly influences SNR! and

Timo O. Korhonen, HUT Communication Laboratory Constellations and eye diagrams Example: 4-ary Pulse Amplitude Modulation (PAM)

Timo O. Korhonen, HUT Communication Laboratory Coherent On-off Keying (OOK) u For on-off keying (OOK) the waveforms are and the optimum coherent receiver can be sketched by

Timo O. Korhonen, HUT Communication Laboratory Phase reversal keying (PRK) (or coherent binary phase shift keying BPSK) u The signaling waveforms are u These waveforms are antipodal (or polar) u Waveform energies are u Therefore BPSK achieves twice the OOK sensitivity or 3 dB improvement. However, it has the same spectral efficiency and is therefore superior to OOK. Also, BPSK is nice in circuit design point of view because it has constant envelope.

Timo O. Korhonen, HUT Communication Laboratory Coherent binary FSK u Assuming frequency difference 2f d the signaling waveforms are now and their energies are u It can be shown that for any choice of modulation index f d /r b u Note that for BPSK the respective energy coefficient is 2E b. Hence BPSK is at least 10log 10 (2/1.22)=2 dB more energy efficient. In addition its circuitry is more simple.

Timo O. Korhonen, HUT Communication Laboratory Simulating BPSK error rate in Matlab

Timo O. Korhonen, HUT Communication Laboratory

transformation of random variables

Timo O. Korhonen, HUT Communication Laboratory Coherent detection: Timing and synchronization u Performance of coherent detection is greatly dependent on how successful local carrier recovery is u Consider bandpass signal s(t) that is applied to the matched filter h(t) with yielding u Therefore, due to phase mismatch at the receiver, the error rate is degraded to

Timo O. Korhonen, HUT Communication Laboratory Example u Assume data rate is 2 kbaud/s and carrier of 100 kHz for a BPSK system. Hence the symbol duration and carrier period are therefore the symbol duration is in radians u Assume carrier phase error is 0.3 % of the symbol duration. Then the resulting carrier phase error is and the error rate for instance for is that should be compared to the error rate without any phase error or u Hence taking care of phase synchronization is very important (or if not possible non-coherent or differentially coherent techniques must be used) (or carrier cycles)

Timo O. Korhonen, HUT Communication Laboratory Noncoherent binary systems: Noisy envelopes u It can be shown that noise plus carrier signal have envelope whose probability distribution function is –For nonzero, constant carrier component A c, Rician distributed: –For zero carrier component Rayleigh distributed:  For large SNR (A C >> s ) the Rician envelope simplifies to  Therefore the received envelope is then essentially Gaussian with the variance  2 and mean equals Distribution centered around A c

Timo O. Korhonen, HUT Communication Laboratory Envelope distribution as the function of carrier component strength Rayleigh distribution Rice distribution

Timo O. Korhonen, HUT Communication Laboratory Noncoherent OOK u Bandpass filter is matched to the signaling waveform, in addition f c >>f m, and therefore the energy for ‘1’ is simply u Envelope distributions follow Rice and Rayleigh for ‘1’ and ‘0’ respectively: distribution for "1" distribution for "0"

Timo O. Korhonen, HUT Communication Laboratory Noncoherent OOK error rate u The optimum threshold is at the intersection of Rice and Rayleigh distributions (areas are the same on both sides) u Usually high SNR is assumed and hence threshold is approximately at the half way and the error rate is the average of '0' and '1' reception probabilities where probability to detect "0" in error probability to detect "1" in error

Timo O. Korhonen, HUT Communication Laboratory Noncoherent FSK u FSK signal consists of two overlapped OOK waveforms on different frequencies: u Therefore noncoherent demodulation of binary FSK is enabled by a pair of bandpass filters:

Timo O. Korhonen, HUT Communication Laboratory Noncoherent FSK error probability u For signal at the ‘1’ branch the envelope distribution is Rician and the other (noise) branch Rayleigh and vice versa u Decision threshold is in the middle and therefore for equal probable ‘1’ and ‘0’ where Y 1 and Y 0 are the decision variables at the output of the respective enveloped detectors u This probability is calculated by first determining the joint PDF p(Y 0,Y 1 ) of Y 1 and Y 0 that is now their product because ‘1’ and ‘0’ are assumed to be independent. Then by using the definition of condition PDF: that yields then the error rate that is about the same as for OOK or u Has constant envelope, does not need local carrier, decision threshold is not a function of channel SNR as in OOK

Timo O. Korhonen, HUT Communication Laboratory Differentially coherent PSK (DPSK) u This methods circumvents using coherent local oscillator but achieves almost the same performance as PSK: u After the multiplier the signal is and the decision variable after the LPF is

Timo O. Korhonen, HUT Communication Laboratory Differential encoding and decoding u Easy data decoding recalls for differential decoding: u Decoding is obtained by the simple rule: that is realized by the circuit shown. u Note that no local oscillator is required u How would you construct the encoder? mkmk akak A B Y XOR

Timo O. Korhonen, HUT Communication Laboratory DPSK error rate u Note that the decision variable has opposite polarity for ‘1’ and ‘0’ and thus it behaves essentially the polar-way u Therefore error rate can be solved analogous to non-coherent FSK but now there exist twice the power of FSK for decision u Therefore DPSK SNR has twice the SNR of non-coherent FSK and hence u Respective error rate penalty to PSK on about 1 dB when compared to coherent PSK when p e <10 -4 u Therefore DPSK is very popular method for instance in modem technology, still it has a tendency to have burst errors due to differential encoding

Timo O. Korhonen, HUT Communication Laboratory BPSK and DPSK error rates by Mathcad ®

Timo O. Korhonen, HUT Communication Laboratory 2.4 kbps DQPSK receiver

Timo O. Korhonen, HUT Communication Laboratory

M-ary PSK systems u The modulated signal consists of the i and q components where with u Symbol energy is comparable to the pure sinusoidal (envelope is not modulated) or u Note that the respective M-level transmission bandwidth for raised cos-pulses is

Timo O. Korhonen, HUT Communication Laboratory

M-ary PSK phase components u M-ary PSK receiver includes the familiar two filters producing respective i- and -q components u Note that the decision variables are u Consider the dashed region for which and assume and therefore

Timo O. Korhonen, HUT Communication Laboratory Error rate  For the transmitted  no error happens if, therefore u The noisy angle is formed by bandpass noise as we already noted or  Under large signal condition, e.g. A C >> , this angle has the distribution that is essentially Gaussian as we noticed earlier (slide 16). For small angle

Timo O. Korhonen, HUT Communication Laboratory u Depending on signal component magnitude A c this distribution varies between Gaussian and uniform distribution: u SNR for a (modulated) constant envelope sinusoid: u Change of variables with

Timo O. Korhonen, HUT Communication Laboratory M-ary PSK error rate expressed by Q-function u The result applies Gaussian CDF or therefore u We derived earlier the case M=2 that is

Timo O. Korhonen, HUT Communication Laboratory M-ary DPSK u If the carrier synchronization is difficult, M-ary DPSK principle can be used yielding u Sensitivity difference of DPSK and PSK is proportional to the fraction of the respective SNR difference or u For M=4 this ratio is 2.3 dB and if M is increased this ration approaches 3 dB

Timo O. Korhonen, HUT Communication Laboratory

Error rate comparison a: Coherent BPSK b: DPSK c:Coherent OOK d: Noncoherent FSK e: noncoherent OOK a: Coherent BPSK b: DPSK c:Coherent OOK d: Noncoherent FSK e: noncoherent OOK

Timo O. Korhonen, HUT Communication Laboratory Summary of M-ary modulation u Quadrature modulation methods yield high spectral and power efficiencies (narrow B T and low error rate) u Coherent modulation yields improved error rates when compared to non- coherent modulation but requires local carrier u Especially rapidly changing propagation delay (statistical multipaths) can make coherent detection very difficult u Differentially coherent is a good compromise: no local carrier but performance almost the same as with the coherent detection u Spectral pre-filtering (as in the Gaussian Minimum Shift Keying, GMSK) can narrow down bandwidth further and the introduced IS is quite small (usually worthwhile to pay the price) u For quantitative comparison assume: u M-ary transmission: fixed symbol rate r of and energy E per symbol u AWGN channel (we inspect the multipath channel later)

Timo O. Korhonen, HUT Communication Laboratory Comparison of quadrature modulation methods (p e =10 -4 ) Note that still the performance is good, envelope is not constant. APK (or MQASK) is used in cable modems APK=MQASK

Timo O. Korhonen, HUT Communication Laboratory