# Carrier phase modulation

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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.

$\forall m, m\in \{1, 2, , M\}\colon {s}_{m}(t)=A{P}_{T}(t)\cos (2\pi {f}_{c}t+\frac{2\pi (m-1)}{M})$

Binary ${s}_{1}(t)$ or ${s}_{2}(t)$

## 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

${E}_{s}=\int_{()} \,d t$ A 2 P T t 2 2 f c t 2 m 1 M 2 t 0 T A 2 1 2 1 2 4 f c t 4 m 1 M
${E}_{s}=\frac{A^{2}T}{2}+\frac{1}{2}A^{2}\int_{0}^{T} \cos (4\pi {f}_{c}t+\frac{4\pi (m-1)}{M})\,d t\approx \frac{A^{2}T}{2}$
(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

${P}_{e}=Q(\sqrt{\frac{2({E}_{s})}{{N}_{0}}})=Q(A\sqrt{\frac{T}{{N}_{0}()}})$
${r}_{t}=(A{P}_{T}(t)\cos (2\pi {f}_{c}t+))+{N}_{t}$
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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