A description of phase shift keying in which information is conveyed through the phase of the modulated signal.
Phase shift keying (psk)
Information is impressed on the phase of the carrier. As data
changes from symbol period to symbol period, the phase shifts.
$\forall m, m\in \{1, 2, , M\}\colon {s}_{m}(t)=A{P}_{T}(t)\cos (2\pi {f}_{c}t+\frac{2\pi (m-1)}{M})$
Representing the signals
An orthonormal basis to represent the signals is
${}_{1}(t)=\frac{1}{\sqrt{{E}_{s}}}A{P}_{T}(t)\cos (2\pi {f}_{c}t)$
${}_{2}(t)=\frac{-1}{\sqrt{{E}_{s}}}A{P}_{T}(t)\sin (2\pi {f}_{c}t)$
The signal
${S}_{m}(t)=A{P}_{T}(t)\cos (2\pi {f}_{c}t+\frac{2\pi (m-1)}{M})$
${S}_{m}(t)=A\cos \left(\frac{2\pi (m-1)}{M}\right){P}_{T}(t)\cos (2\pi {f}_{c}t)-A\sin \left(\frac{2\pi (m-1)}{M}\right){P}_{T}(t)\sin (2\pi {f}_{c}t)$
The signal energy
${E}_{s}=\int_{()} \,d t$∞
∞
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${E}_{s}=\frac{A^{2}T}{2}+\frac{1}{2}A^{2}\int_{0}^{T} \cos (4\pi {f}_{c}t+\frac{4\pi (m-1)}{M})\,d t\approx \frac{A^{2}T}{2}$
(Note that in the above equation, the integral in the last
step before the aproximation is very small.)Therefore,
${}_{1}(t)=\sqrt{\frac{2}{T}}{P}_{T}(t)\cos (2\pi {f}_{c}t)$
${}_{2}(t)=-\sqrt{\frac{2}{T}}{P}_{T}(t)\sin (2\pi {f}_{c}t)$
In general,
$\forall m, m\in \{1, 2, , M\}\colon {s}_{m}(t)=A{P}_{T}(t)\cos (2\pi {f}_{c}t+\frac{2\pi (m-1)}{M})$
and
${}_{1}(t)$
${}_{1}(t)=\sqrt{\frac{2}{T}}{P}_{T}(t)\cos (2\pi {f}_{c}t)$
${}_{2}(t)=\sqrt{\frac{2}{T}}{P}_{T}(t)\sin (2\pi {f}_{c}t)$
${s}_{m}=\left(\begin{array}{c}\sqrt{{E}_{s}}\cos \left(\frac{2\pi (m-1)}{M}\right)\\ \sqrt{{E}_{s}}\sin \left(\frac{2\pi (m-1)}{M}\right)\end{array}\right)$
Demodulation and detection
${r}_{t}={s}_{m}(t)+{N}_{t}\text{for some}m\in \{1, 2, , M\}$
We must note that due to phase offset of the oscillator at the
transmitter,
phase jitter or
phase
changes occur because of propagation delay.
${r}_{t}=A{P}_{T}(t)\cos (2\pi {f}_{c}t+\frac{2\pi (m-1)}{M}+)+{N}_{t}$
For binary PSK, the modulation is antipodal, and the optimum
receiver in AWGN has average bit-error probability
${P}_{e}=Q(\sqrt{\frac{2({E}_{s})}{{N}_{0}}})=Q(A\sqrt{\frac{T}{{N}_{0}()}})$
The receiverwhere
${r}_{t}=(A{P}_{T}(t)\cos (2\pi {f}_{c}t+))+{N}_{t}$
The statistics
${r}_{1}=\int_{0}^{T} {r}_{t}\cos (2\pi {f}_{c}t+\stackrel{}{})\,d t=(\int_{0}^{T} A\cos (2\pi {f}_{c}t+)\cos (2\pi {f}_{c}t+\stackrel{}{})\,d t)+\int_{0}^{T} \cos (2\pi {f}_{c}t+\stackrel{}{}){N}_{t}\,d t$
${r}_{1}=(\frac{A}{2}\int_{0}^{T} \cos (4\pi {f}_{c}t++\stackrel{}{})+\cos (-\stackrel{}{})\,d t)+{}_{1}$
${r}_{1}=(\frac{A}{2}T\cos (-\stackrel{}{}))+\int_{0}^{T} (\frac{A}{2}\cos (4\pi {f}_{c}t++\stackrel{}{}))\,d t+{}_{1}(\frac{AT}{2}\cos (-\stackrel{}{}))+{}_{1}$
where
${}_{1}=\int_{0}^{T} {N}_{t}\cos ({}_{c}t+\stackrel{}{})\,d t$ is zero mean Gaussian with
$\mathrm{variance}\approx \frac{^{2}{N}_{0}T}{4}$ .
Therefore,
$\langle {P}_{e}\rangle =Q(\frac{2\frac{AT}{2}\cos (-\stackrel{}{})}{2\sqrt{\frac{^{2}{N}_{0}T}{4}}})=Q(\cos (-\stackrel{}{})A\sqrt{\frac{T}{{N}_{0}}})$
which is not a function of
$$ and depends strongly on
phase accuracy.
${P}_{e}=Q(\cos (-\stackrel{}{})\sqrt{\frac{2{E}_{s}}{{N}_{0}}})$
The above result implies that the amplitude of the local
oscillator in the correlator structure does not play a role inthe performance of the correlation receiver. However, the
accuracy of the phase does indeed play a major role. Thispoint can be seen in the following example:
${x}_{{t}^{}}=-1^{i}A\cos (-(2\pi {f}_{c}{t}^{})+2\pi {f}_{c})$
${x}_{t}=-1^{i}A\cos (2\pi {f}_{c}t-2\pi {f}_{c}{}^{}-2\pi {f}_{c}+{}^{})$
Local oscillator should match to phase
$$ .
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