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Table of Contents
ELEN90057 - Communication Systems
Analogue communication
Bandpass spectra
Often we are interested in spectra confined to $\pm0.5B\text{ Hz}$ of a centre frequency $f_0$, being a band. If $f_0>>B$, this is a narrow band. For real signals and systems, the spectrum for the negative frequencies is redundant as $S(f)=S^*(-f)$. The signal can be in bandpass form, that is a modulation of a sinusoid of the carrier frequency. We want to work with the interesting signal, not the carrier, so we transform the bandpass to a lowpass representation. First we remove the negative frequencies, providing the envelope: $$Z(f)=(1+\text{sign}(f))S(f)=\begin{cases}2S(f),&f>0\\S(0),&f=0\\0,&f<0\end{cases}$$ Then we down-shift in the frequency domain, providing the lowpass highpass spectrum: $$S_l(f)=Z(f+f_0)$$ Note that in general $S_l(f)\neq S_l^*(-f)$, being that it is not a real signal in time domain ($s_l(t)$).
To convert from the lowpass to bandpass representation, we do the inverse. First we shift the spectrum up: $$Z(f)=S_l(f-f_0)$$ Then we reflect around 0 Hz, conjugate, add to the pre-envelope and scale by 0.5: $$S(f)=\frac{Z(f)+Z^*(-f)}{2}=\frac{S_l(f-f_0)+S_l^*(-f-f_0)}{2}$$
In time domain, the transformation is a bit more tricky. First, we note that $\text{sign}(t)\iff\frac{1}{j\pi f}\therefore\text{sign}(f)\iff\frac{-1}{j\pi t}=\frac{j}{\pi t}$ (this is taking the Hilbert Transform $H(f)=-j\text{sign}(f)$). We can find $z(t)$ from the IFT of $Z(f)$: $$z(t)=s(t)+\frac{j}{\pi t}*s(t)=s(t)+\frac{j}{\pi}\int_{-\infty}^\infty\frac{s(\tau)}{t-\tau}=s(t)+j\hat{s}(t)$$ Where $\hat{s}=s*\frac{1}{\pi t}$. Then we take the IFT of $Z(f)=S_l(f-f_0)$: $$s_l(t)=e^{-j2\pi f_0t}z_0(t)\implies z(t)=e^{j2\pi f_0t}s_l(t)$$ Lowpass to bandpass is: $$s(t)=\mathcal{R}(z(t))=\mathcal{R}(e^{j2\pi f_0t}s_l(t))$$ This lets us write $s(t)=|s_l(t)|e^{j\angle s_l(t)}=A(t)e^{j\phi(t)}$. As a lowpass, $A(t)$, $\phi(t)$ change slowly with time. $$z(t)=A(t)e^{j(2\pi f_ct+\phi(t)}$$ The phasor rotates slowly at angular velocity $\approx 2\pi f_c$, with slowly varying magnitude and phase. The real part is: $$s(t)=A(t)\cos(2\pi f_ct+\phi(t))$$ Any real bandpass signal can be represented in this way. As $s_l(t)=s_c+js_s$, the real part is the in-phase component and the complex is the quadrature. We can find that: $$s(t)=s_c(t)\cos(2\pi f_0t)-s_s(t)\sin(2\pi f_0t)$$ This is the canonical form of a bandpass signal, whereas $s_l(t)e^{j2\pi f_0t}$ is the polar form. We can find that: $$s_c(t)=s(t)\cos(2\pi f_0t)+\hat{s}(t)\sin(2\pi f_0t)$$ $$s_s(t)=-s(t)\sin(2\pi f_0t)+\hat{s}(t)\cos(2\pi f_0t)$$
The bandpass response of the Hilbert transform is $\frac{1}{\pi t}$, so $H(f)=-j\text{sign}(f)$. The amplitude response is unchanged, but the positive phase is less $\pi/2$ and the negative phase is added $\pi/2$. Taking the lowpass representation removes the high frequency carrier signal. $$A(t)\cos(2\pi f_0t+\phi(t))\iff A(t)e^{j\phi(t)}$$
Double side band suppressed carrier modulation
Here, $m(t),M(f)$ is the message signal. To up-convert, we multiply a message by a carrier wave $A_c\cos(2\pi f_ct)$. $$s(t)=m(t)A_c\cos(2\pi f_ct)=0.5A_cm(t)e^{j2\pi f_ct}+0.5A_cm(t)e^{-j2\pi f_ct}$$ $$S(f)=0.5A_CM(f-f_c)+0.5A_CM(f+f_c)$$ DSBSC modulation shifts the message spectrum left and right by $\pm f_c$. This creates a bandpass signal. We can then find the low pass representation: $$s_l(t)=A_cm(t)$$ This is a real signal, whereas in general the lowpass is a complex number. This suggests that we could send another signal in the complex, quadrature component. This lets us send two separate messages at the same time, called quadrature carrier multiplexing. $$s_c(t)=A_cm_1(t)$$ $$s_s(t)=A_cm_2(t)$$ This requires using a balanced modulator (mixer or multiplier) to combine the carrier and the message. The second message has its carrier shifted by $90^\circ$. The resultant signal is: $$s(t)=A_cm_1(t)\cos(2\pi f_ct)+A_cm_2(t)\sin(2\pi f_ct)$$ $$s_c=A_cm_1(t),s_s=-A_cm_2(t)$$ To recover these signals at the receiving end, a Phase Locked Loop is required to recover the carrier from the signal. This recovered signal is then fed through a balanced modulator with the received signal, and the message is recovered after the modulator output is fed through a lowpass filter.
Amplitude modulation
Suppose we transmit: $$s(t)=A_c(1+\mu m(t))\cos(2\pi f_ct)$$ We choose amplitude sensitivity or modulation index $\mu$ such that $$|m(t)|<1/\mu\implies1+\mu m(t)>1-\mu|m(t)|>0,\forall t$$ This is that $1=\mu m(t)$ is the envelope of $s(t)$. Further we choose $f_c>>W$ (message BW), so that $m(t)\approx c$, a constant. Over each carrier cycle, the maxima of $s(t)$ follows the envelope.
A phase reversal is related to the change in the sign of the amplitude. When $A(t)<0$, $s(t)=-|A(t)|\cos(2\pi f_ct)=|A(t)|\cos(2\pi f_ct+\pi)$, causing a bump in the signal from the sudden change in phase. $A(t)$ needs to stay positive all the time. When $\mu<\frac{1}{|m(t)|}$, meaning $1-\mu|m(t)|>0$, and as $m(t)\geq -|m(t)|$, $1+\mu m(t)\geq 1\mu|m(t)|>0$, resulting in the amplitude always being positive. This removes phase reversals, making envelope detection easier.
Envelope detection tries to remove the high frequency carrier to find the message. It results in finding the magnitude of the signal. Envelope detection recovers the signal without needing a PLL to regenerate the carrier, unlike Quadrature-carrier multiplexing. This is also called asynchronous detection, incoherent detection or direct detection.
Side bands
Side bands are the parts of the signal outside 0 in the passband representation. The upper side band is to the right, and the lower is to the left. The upper and lower side bands are related as: $$USB=LSB^*$$ When we convert to a bandpass signal, we are centred around $f_c$, but still have upper and lower side bands flanking $f_c$. In a DSBSC signal, there is four-fold redundancy, as there are two copies of the LSB and of the USB. As such, we can transmit only one of the side bands and still receive the whole signal. This single side band representation is denoted with as $\tilde{m}(t)$ and $\tilde{M}(f)$. The signal is: $$\tilde{M}(f)=(1+\text{sign}(f))M(f)=\begin{cases}2M(f),&f>0\\0,&f<0\end{cases}$$ $$S_{SSB}(f)=0.5A_C\tilde{M}(f-f_c)+0.5A_C\tilde{M}(-f-f_c)=\begin{cases}A_CM(f-f_c),&f>f_c\\A_CM(f+f_c),&f<f_c\\0,&\text{elsewhere}\end{cases}$$ In time domain this corresponds to: $$\tilde{m}(t)=m(t)+j\hat{m}(t)$$ $$s_{SSB}(t)=A_Cm(t)\cos(2\pi f_ct)-A_C\hat{m}(t)\sin(2\pi f_ct)=\mathcal{R}\{\tilde{m}(t)A_Ce^{j2\pi f_ct}\}$$ $$s_C=A_Cm(t),s_S=A_C\hat{m}(t)$$ This is the same as quadrature carrier multiplexing, with $m_2=\hat{m}$.
Digital communications
Digital signal are encoded and modulated before being transmitted over a channel, demodulated and decoded. The modulator turns the digital signal into a time domain signal. A communication channel can include space, atmosphere, optical disks, cables, etc. Different channels cause different types of impairments and require different types of modulation.
Modulation involves mapping the digital information into analogue signals for transmission over physical channels. It requires parsing of the incoming bit sequence into a sequence of binary words length $k$. Each binary word corresponds to a symbol, with $M=2^k$ possible symbols. Each symbol has a signalling interval of length $T$. $1/T$ is the symbol rate, with $k/T$ being the bit rate.
A baseband signal is a low frequency real signal centred on 0. A bandpass signal is a high frequency signal centred on $f_C$. There is no need to use a carrier waveform to transmit a baseband signal. A bandpass signal needs passbands far from 0 to be removed.
Signal space
To simplify analysis, geometric vector representation is used for baseband and bandpass signals. A vector space or linear space $L$ of a field $F$ (usually $\mathbb{R}$ or $\mathbb{C}$) is a set that is closed over addition and scalar multiplication and is:
- Associative
- Commutative
- Distributive
- Has additive identity
- Has multiplicative identity
Signal space is a vector space consisting of functions $x(t)$ defined on a time set $T$. The modulation scheme is visualised as a finite set of points, called the signal space diagram or signal constellation. This enables a geometric interpretation, and allows us to treat bandpass modulation similarly to baseband modulation.
The inner product of two complex valued signals is: $$\langle x_1(t),x_2(t)\rangle=\int_{-\infty}^\infty x_1(t)x^*_2(t)dt$$ Two signals are orthogonal if $\langle x_1(t),x_2(t)\rangle=0$. The norm of $x(t)$ is: $$||x(t)||=\sqrt{\langle x(t),x(t)\rangle}=\sqrt{\int_{-\infty}^\infty|x(t)|^2dt}=\sqrt{\varepsilon_x}$$ Where $\varepsilon_x$ is the energy in $x(t)$. The distance between two signals is: $$d(x_1(t),x_2(t))=||x_1(t)-x_2(t)||$$ The Cauchy-Schwartz inequality for two signals is: $$|\langle x_1(t),x_2(t)\rangle|\leq||x_1(t)||\cdot||x_2(t)||=\sqrt{\varepsilon_{x_1}\varepsilon_{x_2}}$$ With equality when $x_1(t)=\alpha x_2(t)$, where $\alpha$ is any complex number. The triangle inequality is: $$||x_1(t)+x_2(t)||\leq||x_1(t)||+||x_2(t)||$$
A set of $N$ signals $\{\phi_j(t),j=1,2,...,N\}$ spans a subspace $S$ if any signal can be written as a linear combination of the $N$ signals. $$s(t)=\sum_{j=1}^N s_j\phi_j(t)$$ Where $s_j$ are scalar coefficients. A set of signals is linearly independent if no signal in the set can be represented as a linear combination of the other signals in the set. A basis for $S$ is any linearly independent set that spans the whole space. The dimension of $S$ is the number of elements in any basis for $S$. An orthonormal basis is a basis such that: $$\langle \phi_j(t),\phi_n(t)\rangle=\int_{-\infty}^\infty \phi_j(t)\phi_n^*(t)dt=\begin{cases}1,j=n\\0,j\neq n\end{cases}$$ Orthonormal bases provide convenient ways of representing any set. Using an orthonormal basis allows us to easily express a signal as a point in signal space. Each point using a basis of $M$ dimensions corresponds to $k=\log_2M$ bits of information. The square of the Euclidean distance of a point to the origin equals the energy of the corresponding signal: $$\varepsilon_{s_m}=s_{m1}^2+s_{m2}^2+...+s_{mN}^2$$ If a given signal $r(t)$ is outside of the subspace, we can project the signal onto the space to get $\hat{s}(t)$. The Gram-Schmidt procedure can be used to construct an orthonormal basis, by iteratively projecting vectors onto vectors in the basis and removing the projection from the original vector. This can always be used to construct an orthonormal basis.
Digital Modulation
For Pulse Amplitude Modulation (PAM), we turn each block into a distinct amplitude. This uses a signal generator to map the sequence of blocks of length $k$ into the sequence of $M=2^k$ possible symbols. We then use a modulator to map the symbol sequence to a continuous time signal.
One-dimensional modulation provides one of the simplest modulation schemes, One Off Keyring (OOK). A baseband OOK modulator maps a binary symbol sequence $a(n)$ to continuous time signal $s(t)$ by: $$s(t)=\sum_{n\in\mathbb{Z}} a(n)p(t-nT)$$ Where $1/T$ is the symbol rate and $p(t)$ is a pulse signal. There are various ways of encoding the bits, such as non-return-to-zero (NRZ), return-to-zero (RZ), Manchester (MAN) and half-sine (HS), each with distinct pulse shapes. Baseband OOK modulation produces a signal of the form: $$s_m(t)=A_mp(t);m=1,2;A_1=1,A_2=0$$ Bandpass OOK employs a carrier frequency $f_c$ to produce a signal of the form: $$s_m(t)=A_mg(t)\cos(2\pi f_ct);m=1,2;A_1=1,A_2=0$$ Where the pulse signal is denoted by $g(t)$. This is the same as DSBSC modulation. The constellation is given by points at $(0,0)$ and $(1,0)$.
A baseband PAM modulator maps a symbol sequence maps a signal $a(n)$ to a continuous time signal $s(t)$: $$s(t)=\sum_{n\in\mathbb{Z}}a(n)p(t-nT)$$ This uses amplitudes above and below 0, at many different levels. It produces a constellation on the x-axis mirrored on the y-axis. The mapping on the constellation uses grey coding for adjacent points. Grey coding minimises the errors in transmission.
When choosing the signals, we want the to be as far apart from each other when drawn as a constellation. The energy for a constellation point is: $$\varepsilon_m=||s_m(t)||^2=\int_0^TA_m^2p^2(t)dt=A^2\varepsilon_p$$ Where $\varepsilon_p$ is the energy in $p(t)$ and $m=1,...,M$. An orthonormal basis vector for PAM is given by: $$\phi(t)=\frac{p(t)}{\sqrt{\varepsilon_p}}$$ For carrier modulated PAM signals, we have: $$s_m(t)=A_mg(t)\cos(2\pi f_ct),1\leq m\leq M, 0\leq t<T$$ The bandpass PAM energy for a constellation point equals: $$\varepsilon_m=||s_m(t)||^2=\frac{A_m^2}{2}\int_0^Tg^2(t)dt+\frac{A_m^2}{2}\int_0^Tg^2(t)\cos(4\pi f_ct)dt\approx\frac{A_m^2}{2}\varepsilon_g$$ Binary bandpass PAM is also called Binary Phase Shift Keyring (BPSK) because the symbol values inform the phase of $s_m(t)$. Modulation of the signal waveform $s_m(t)$ with carrier $\cos(2\pi f_c t)$ shifts the spectrum of the baseband signal by $f_c$: $$S_m(f)=\frac{A_m}{2}(G_T(f-f_c)+G_T(f+g_c))$$ For bandpass PAM signalling, the orthonormal basis vector is given by: $$\phi(t)=\sqrt{\frac{2}{\varepsilon_p}}g(t)\cos(2\pi f_ct)$$ Which results in $s_m(t)=A_m\sqrt{\frac{\varepsilon_g}{2}}\phi(t)$. Bandpass PAM has the same signal space diagram as baseband PAM, but with a different basis vector.
The modulator is implemented by feeding the input bits into a serial to parallel converter, outputting $\log_2M$ bits. This is fed into a Look Up Table (LUT) to find the symbols, which is fed into a pulse shaping filter to generate the signal. The signal may be up-sampled in filtering (FIR possibly) and fed into a DAC to create the analogue form.
Two dimensional modulation
Orthogonal signalling involves modulation using two signals that are orthogonal: $$s(t)=s_1\phi_1(t)+s_2\phi_2(t)$$ We can denote the modulated signals as a vector $s(t)=(s_1,s_2)$. Signals with M-PSK modulation can be represented as: $$s_m(t)=g(t)\cos(2\pi f_ct+\theta_m)=\mathcal{R}(g(t)e^{j\theta_m}e^{j2\\pi f_ct})=g(t)\cos(\theta_m)\cos(2\pi f_ct)-g(t)\sin(\theta_m)\sin(2\pi f_ct)$$ Where $g(t)$ is the signal pulse shape and $\theta_m=\frac{2\pi}{M}(m-1)$ is the phase the conveys the transmitted information. An orthonormal basis for the signal space is: $$\{\phi_1(t),\phi_2(t)\}=\left\{\sqrt{\frac{2}{\varepsilon_g}}g(t)\cos(2\pi f_ct),-\sqrt{\frac{2}{\varepsilon_g}}g(t)\sin(2\pi f_ct)\right\}$$ Where the basis functions are unit norm, $||\phi_1(t)||=||\phi_2(t)||=1$. In M-ary phase-shift keyring (PSK), all M bandpass signals are constrained to have the same energy (signal constellation points lie on a circle). Grey encoding is used so that adjacent phases differ by only one bit. This leads to a better average bit error rate (BER). The transmitted information is impressed on 2 orthogonal carrier signals, the in-phase carrier $\cos(2\pi f_ct)$ and the quadrature carrier $\sin(2\pi f_ct)$. $$s_m(t)=g(t)\cos(\theta_m)\cos(2\pi f_ct)-g(t)\sin(\theta_m)\sin(2\pi f_ct)$$ Its lowpass equivalent signal is: $$s_m^{lowpass}(t)=g(t)e^{j\Theta_m}=I(t)+jQ(t)$$ Where $I(t)=g(t)\cos(\theta_m)$ is the in-phase component and $Q(t)=g(t)\sin(\theta_m)$ is the quadrature component.
In quadrature/quaternary PSK (QPSK), the $M=4$ signal points are differentiated by phase shifts in multiples of $\pi/2$. $$s_m(t)=g(t)\cos\left(2\pi f_ct+\frac{\pi}{2}(m-1)\right)$$ Equivalently the signal constellation can be rotated so that the vectors are in the quadrants rather than on the axes. $$s(t)=I(t)\sqrt{2}\cos(2\pi f_ct)-Q(t)\sqrt{2}\sin(2\pi f_c(t))$$ $I(t)=\sum_na_1(n)f(t-nT)$ is the in-phase component of $s(t)$ and $Q(t)=\sum_na_2(n)g(t-nT)$ is the quadrature component of $s(t)$. $I(t)$ and $Q(t)$ are binary PAM pulse trains, making QPSK interpretable as two PAM signal constellations on orthogonal axes.
Two dimensional bandpass modulation
Quadrature amplitude modulation (QAM) allows signals to have different amplitudes and impress separate information bits on each of the quadrature carriers. Important performance parameters are the average energy and minimum distance in the signal constellation. The signal constellation consists of concentric circles. The points are all off axis in each quadrant for maximum efficiency (minimises energy and maximises minimum distance).
Comparison
| Scheme | $s_m(t)$ | $s_m$ | $E_{avg}$ | $E_{bavg}$ | $d_{min}$ |
| Baseband PAM | $A_mp(t)$ | $A_m\sqrt{\varepsilon_p}$ | $\frac{2(M^2-1)}{3}\varepsilon_p$ | $\frac{2(M^2-1)}{3\log_2M}\varepsilon_p$ | $\sqrt{\frac{6\log_2M}{M^2-1}\varepsilon_{bavg}}$ |
| Bandpass PAM | $A_mg(t)\cos(2\pi f_ct)$ | $A_m\sqrt{\frac{\varepsilon_p}{2}}$ | $\frac{M^2-1}{3}\varepsilon_p$ | $\frac{M^2-1}{3\log_2M}\varepsilon_p$ | $\sqrt{\frac{6\log_2M}{M^2-1}\varepsilon_{bavg}}$ |
| PSK | $g(t)\cos\left[2\pi f_ct+\frac{2\pi}{M}(m-1)\right]$ | $\sqrt{\frac{\varepsilon_g}{2}}\left(\cos\frac{2\pi}{M}(m-1),\sin\frac{2\pi}{M}(m-1)\right)$ | $\frac{1}{2}\varepsilon_g$ | $\frac{1}{2\log_2M}\varepsilon_g$ | $2\sqrt{\log_2M\sin^2\left(\frac{\pi}{M}\right)\varepsilon_{bavg}}$ |
| QAM | $A_{mi}g(t)\cos(2\pi f_ct)-A_{mq}g(t)\sin(2\pi f_ct)$ | $\sqrt{\frac{\varepsilon_g}{2}}(A_{mi}\cdot A_{mq})$ | $\frac{M-1}{3}\varepsilon_g$ | $\frac{M-1}{3\log_2M}\varepsilon_g$ | $\sqrt{\frac{6\log_2M}{M-1}\varepsilon_{bavg}}$ |
Multidimensional modulation
We can use time domain and/or frequency domain to increase the number of dimensions. Orthogonal Signalling (Baseband) is one example, e.g. Pule position modulation (PPM). This varies where in the pulse position within the pulse time, i.e. first quarter of pulse or third quarter. Alternatively we can frequency shift (FSK) to create orthogonal signals: $$s_m(t)=\sqrt{\frac{3\varepsilon}{T}}\cos(2\pi(f_c+m\Delta f)t), 0\leq m\leq M-1,0\leq t\leq T$$ For orthogonality, we need a minimal frequency separation of $\Delta f=1/(2T)$.
Receiving signals
Additive White Gaussian Noise
A received signal has Additive White Gaussian Noise added to it from the channel. $$r(t)=s_m(t)+n(t)$$ Where $n(t)$ is a AWGN random noise process with power spectral density $\frac{N_0}{2}$ W/Hz. Given $r(t)$, a receiver must decide which $s_m(t)$ was transmitted, minimising the error probability, or other criteria.
The $Q$ function is useful for tail probabilities, and is defined as: $$Q(x)=P[X>x]=\frac{1}{\sqrt{2\pi}}\int_x^\infty\exp\left(-\frac{t^2}{2}\right)dt$$ For a general Gaussian RV $X\sim\mathcal{N}(\mu,\sigma^2)$, we know that $\frac{X-\mu}{\sigma}\sim\mathcal{N}(0,1)$, so: $$P[X>x]=Q(\frac{x-\mu}{\sigma})$$ A continuous time random process $X(t)$ is completely characterised bu joint PDFs of the form: $$f_{X(t_1),X(t_2),...,X(t_n)}(x_1,x_2,...,x_n)$$ $X(t)$ is strictly stationary if for all $n,\Delta,(t_1,t_2,...,t_n)$, we have: $$f_{X(t_1),X(t_2),...,X(t_n)}(x_1,x_2,...,x_n)=f_{X(t_1+\Delta),X(t_2+\Delta),...,X(t_n+\Delta)}(x_1,x_2,...,x_n)$$ Random processes have a mean and autocorrelation: $$m_X(t)=E[X(t)]$$ $$R_X(t_1,t_2)=E[X(t_1)X^*(t_2)]$$ A random process is Wide Sense Stationary (WSS) if its mean is constant for all time and its autocorrelation depends only on the time difference $\tau=t_1-t_2$, allowing us to write the autocorrelation as $R_X(\tau)$. The Power Spectral Density (PSD) of a WSS process is the fourier transform of the autocorrelation $\mathcal{S}_X(f)=\mathcal{F}[F_X(\tau)]$. The total power content of the process is: $$P_X=E[|X(t)|^2]=R_X(0)=\int_{-\infty}^\infty\mathcal{S}_X(f)df$$
If a WSS process $X(t)$ passes through a LTI system with impulse response $h(t)$ and frequency response $H(f)$, the output $Y(t)=\int_{-\infty}^\infty X(\tau)h(t-\tau)d\tau$ is also a WSS process. The output process has a mean of $m_Y=m_x\int_{-\infty}^\infty h(t)dt=m_XH(0)$, autocorrelation $R_Y=R_X\star h\star \tilde{h}$ ($\tilde{h}(t)=h(-t)$) and PSD $\mathcal{S}_Y(f)=\mathcal{S}_X(f)|H(f)|^2$. $X(t)$ is a Gaussian random process if $\{X(t_1),X(t_2),...,X(t_n)\}$ has a jointly Gaussian PDF, making $X(t_k)$ a Gaussian RV for any fixed $t_k\in\mathbb{R}$. $X(t)$ is a white noise process if its PSD $S_X(f)$ is constant for all frequencies. The power content of white noise processes $P_X=\int_{-\infty}^\infty\mathcal{S}_X(f)=\infty$, so not physically realisable. A Gaussian process into an LTI system produces a Gaussian process, but a white input does not necessarily produce a white output.
The AWGN process can be modelled as a random process that is:
- WSS, with $R_N(\tau)=\mathcal{F}^{-1}[\mathcal{S}_N(f)]=\frac{N_0}{2}\delta(\tau)$
- Zero mean
- Gaussian, with $N(t)\sim\mathcal{N}(0,\frac{N_0}{2})$
- White, with $\mathcal{S}_N(f)=\frac{N_0}{2}$
Demodulation
The first thing a receiver does is project the received waveform onto a signal $\mathbf{r}=(r_1,r_2,...,r_N)$ in the signal space. The detector then decides which of the possible signal waveforms was transmitted. Any signal can be written as: $$s_m(t)=\sum_{k=1}^Ns_{mk}\phi_k(t)$$ This is added to noise to produce the received signal, which is projected into the signal space. In projecting the signal, we get:
$$r_{mj}=\langle r_m(t),\phi_j\rangle=s_{mj}+n_j$$ Assuming AWGN, $n_j\sim\mathcal{N}(0,\frac{N_0}{2})$, and $E\{n_in_j\}=\delta_{ij}=\frac{N_0}{2}$.
There are two main approaches to demodulating $r(t)$:
- Correlation-type The incoming signal is modulated with the basis signals in parallel, then integrated to find each $r_k$.
- Matched-type The received signal is convoluted with the basis signals in parallel to produce each $r_k$, $r_k(t)=\int_0^tr(t)\phi_k(T-(t-\tau))d\tau=\int_0^tr(\tau)\phi_k(\tau)d\tau$.
Both of these produce signals that are equal at integer multiples of $T_s$.
A correlation-type demolulator uses a parallel bank of $N$ correlators with multiplies $r(t)$ with $\{\phi_k(t)\}_{k=1}^N$. The output is: $$r_k=\int_0^Tr(t)\phi_k(t)dt=s_{mk}+n_k$$ This makes the overall result: $$\mathbf{r}=\mathbf{s}_m+\mathbf{n}$$ The received signal can be expressed as: $$r(t)=\sum_{k=1}^Ns_{mk}\psi_k(t)+\sum_{k=1}^Nn_k\phi_k(t)+n'(t)=\sum_{k-1}^Nr_k\phi_k(t)+n'(t)$$ We ignore $n'(t)=n(t)-\sum_{k=1}^Nn_k\phi_k(t)$ because RV $n'(t)$ and $r_k(t)$ are independent.
A matched filter-type demodulator uses a parallel bank of $N$ linear filters with impulse response $h_k(t)=\phi_k(T-t)$. The output is: $$r_k(t)=(r\star h_k)(t)=\int_0^Tr(\tau)\phi(T-t+\tau)d\tau$$ Hence, $r_k(t)$ is a Gaussian process, which when sampling at $t=T$, we get: $$r_k=\int_0^Tr(\tau)\phi_k(\tau)d\tau$$ A matched filter to a signal $s(t)$ is a filter whose impulse response is $h(t)=s(T-t)$, over the confined time interval $0\leq t\leq T$. A matched filter maximises the signal to noise ratio for a signal corrupted by AWGN. The output SNR from the filter depends on the energy of the waveform $s(t)$ but not on the detailed characteristics of $s(t)$. Sampling the output at time $t=T$ gives the signal and noise components: $$y(t)=\underbrace{\int_0^Ts(\tau)h(T_\tau)d\tau}_{y_s(T)}+\underbrace{\int_0^Tn(\tau)h(T_\tau)d\tau}_{y_n(T)}$$ We need to choose a $h(t)$ to maximise $\frac{y_s^2(T)}{E[y_n^2(T)]}$, being $h(t)=ks(T-t)$ for some constant $k$. This means that we need a filter response that is matched to the signal. The output Signal to Noise Ratio is $\frac{2\varepsilon_s}{N_0}$, depending only on the signal energy.
Detection
The projected signal is mapped to a point in signal space and forms spherical noise clouds due to the Gaussian components. In choosing which signal to map to, we need to chose the closest point. We want to minimise the overall probability of error. This is done by partitioning the signal space into $M$ nonoverlapping regions $D_1,D_2,...,D_M$. The detector design is in choosing these partitions.
The prior probability is the probability that a signal was transmitted ($P(s_m\text{ transmitted})$), the posterior is the probability that a signal was transmitted given what was received ($P(s_m\text{ transmitted}|r\text{ received})$). The likelihood function forms a conditional PDF for $P(r\text{ received}|s_m\text{ transmitted})$. These are all related with Bayes' theorem: $$P(s_m|r)=\frac{P(r|s_m)P(s_m)}{P(r)}$$ We should note that: $$P(\text{No error}|s_m)=P(r\in D_m|s_m)=\int_{D_m}P(r|s_m)dr$$ Minimising the probability of error means maximising the probability of no error. As such, we want to construct the decision regions such that: $$D_m=\{r\in\mathbb{R}^N:P(s_m|r)\geq P(s_{m'}|r),\forall m'\neq m\}$$
The Maximum A Posteriori (MAP) criterion for this is: $$\hat{m}=\arg\max_mP(r|s_m)P(s_m)$$ The MAP detector is an optimal detector for minimising the probability of error.
The Maximum Likelihood (ML) criterion is: $$\hat{m}=\arg\max_mP(r|s_m)$$ The ML detector is optimal when all signals are equiprobable (MAP devolves to ML in such a case).
In AWGN channels, the received vector components of $r=(r_1,r_2,...,r_n)$ are: $$r_k=s_{mk}+n_k$$ Were $n_k\sim\mathcal(0,N_0/2)$ such that $r_i\sim\mathcal{N}(s_{mk},N_0/2)$. The likelihood function can be calculated to be: $$P(r|s_m)=\prod_{k=1}^NP(r_k|s_m)=\frac{1}{(\pi N_0)^{\frac{N}{2}}}\exp\left(-\frac{||r-s_m||^2}{N_0}\right)$$ The ML detection criterion for AWGN channels is given by: $$\hat{m}=\arg\max_m\frac{1}{(\pi N_0)^{\frac{N}{2}}}\exp\left(-\frac{||r-s_m||^2}{N_0}\right)$$ This can be simplified to: $$\hat{m}=\arg\min_m||r-s_m||$$ This means we decide on $s_m$ that is closest to $r$, being minimum distance detection. The decision regions are represented graphically as the midlines between $s_1,s_2,...,s_M$. We can expand the distance metric for ML to: $$||r-s_m||^2=\underbrace{||r||^2}_{\text{Independent of }m}-2\langle r,s_m\rangle+\underbrace{||s_m||^2}_{\text{Energy of m-th signal, }\varepsilon_m}$$ This allows us to reexpress the ML criterion as: $$\hat{m}=\arg\max_m[\langle r,s_m\rangle+\eta_m]$$ Where $\eta=-\frac{1}{2}\varepsilon_m$ is a bias term compensating for signal sets that have unequal energies such as PAM.
Phase mismatch
When the phase of the carrier and receiver are not synchronised, or the carrier frequency is only approximately known there is a mismatch affecting the signal. This mismatch causes: $$r=\pm\sqrt{\frac{2}{\varepsilon_g}}\int_0^Tg^2(t)\cos(2\pi f_ct+\theta)\cos(2\pi f_ct)dt+n\approx\pm\sqrt{\frac{\varepsilon_g}{2}}\cos(\theta)+n$$ When there is a phase mismatch ($\cos(\theta)<1$), there is a loss of information. The phase mismatch causes a rotation that may lead to the projection lying in the wrong region, even in the noiseless case.
If the phase mismatch is unknown and changing rapidly, and if small, it is ingnored and treated as random noise and an otherwise optimal detector is designed. If the mismatch is large then noncoherent demodulators are used. These work best for modulation schemes that ignore phase, such as envelope detectors for orthogonal signalling (FSK) and on-off keyring (OOK). Alternatively is the phase mismatch is unknown but fixed or varying slowly, then using differential modulation that modulate the phase differences rather than absolute phase (e.g. DBPSK).
Non-coherent OOK Demodulation
The transmit signal for OOK is: $$s_m(t)=A_mg(t)\cos(2\pi f_ct);m=1,2;A_1=1,A_2=0$$ The received signal, after AWGN with imperfect synchronisation, is: $$r(t)=A_mg(t)\cos(2\pi f_ct+\theta)+n(t)=A_mg(t)\cos(\theta)\cos(2\pi f_ct)-A_mg(t)\sin(\theta)\sin(2\pi f_ct)+n(t)$$ The dimension of the signal space is 1, but the dimension of the received signal space is 2 due to the phase mismatch. The signal demodulator must have 2 correlators, otherwise some symbol information will be lost. The signal demodulator output is: $$r_1=\sqrt{\frac{1}{\varepsilon_g}}\int_0^\infty A_mg^2(t)\cos(2\pi f_ct+\theta)\cos(2\pi f_ct)dt+n_1\approx\sqrt{\frac{\varepsilon_g}{2}}A_m\cos(\theta)+n_1$$ $$r_2\approx\sqrt{\frac{\varepsilon_g}{2}}A_m\sin(\theta)+n_2$$ This rotates the signal along the signal space circle of radius $\sqrt{\frac{\varepsilon_g}{2}}$. This requires a circular decision region.
The optimal detector uses: $$\hat{m}=\arg\max_{m=1,2}P(s_m)P(r|s_m)=\arg\max_{m=1,2}P(s_m)\int_0^{2\pi}P(r|s_m,\theta)P(\theta)d\theta$$ The worst case has phase uncertainty uniformly distributed over $[0,2\pi)$. Assuming equiprobable symbols, we get: $$\hat{m}=\begin{cases}1,&r_1^2+r_2^2>V_T\\2,&r_1^2+r_2^2<V_T\end{cases}$$ Which depend only on the envelope of the properly received signal ($V_T$ in terms of Bessel function). At high SNRs: $$r_1^2+r_2^2\approx\frac{1}{2}A_m^2\varepsilon_g(\cos^2(\theta)+\sin^2(\theta))=\begin{cases}\frac{1}{2}\varepsilon_g,&m=1\\0,&m=2\end{cases}$$ Which is independent of $\theta$.
Non-coherent FSK demodulation
Similarly to non-coherent OOK, we can demodulate Binary FSK. That is we demodulate against sin and cos, and use the squares to make a decision with a larger envelope. This can be extended to M-ary FSK using a larger envelope still.
Differential Modulation
Differential MPSK modulation involves precoding of the infomation symbol sequence $b(n)$ into a symbol sequence $\delta(n)$ that is then input to a MPSK modulator. The DMPSK demodulator needs to invert the precoding. In MPSK each symbol value determines the actual value of the phase, but in DMPSK it determines the phase change form the previous signalling interval's period. This makes DMPSK modulation with memory. Differential demodulation is used in situations where the MPSK demodulator's errors are always of the same type, determined by a fixed (or slowly varying) phase mismatch $\varphi$, the value of which we do not know (may be zero). Differential modulation leads to a better symbol error performance since it is robust against a fixed unknown phase mismatch. If the demodulator makes an error, there are bit errors over that bit and the next one, due to the differential coding nature.
Error probability
No matter the demodulation, as we are using probabilistic determination with noise, there exists a chance we can make an error. This is when we decide on the received signal being misidentified as another.
For Binary PAM, the probability we make an error is: $$P(err|s_2)=P(r>0,s_2)$$ We can recall that $r|s_2\sim\mathcal{N}\left(-\sqrt{\varepsilon_b},\frac{N_0}{2}\right)$, making the probability of an error equal to: $$P(err|s_2)=Q\left(\frac{0+\sqrt{\varepsilon_b}}{\sqrt{\frac{N_0}{2}}}\right)=Q\left(\sqrt{\frac{2\varepsilon_b}{N_0}}\right)$$ Due to symmetry, the error is the same for $s_1$. The bit error probability is: $$P_b=\frac{1}{2}P(err|s_1)+\frac{1}{2}P(err|s_2)=Q\left(\sqrt{\frac{2\varepsilon_b}{N_0}}\right)$$
For binary orthogonal signals, the signal is: $$s_m(t)=\sqrt{\varepsilon_b}\phi_m(t)$$ Where $\phi_1(t),\phi_2(t)$ are orthonormal. The ML detector uses: $$\hat{m}=\arg\min_m||r-s_m||$$ Being that the constellation point closest to the received point is assumed to be transmitted. The error probability consists of 2 normally distributed variables, unlike the one from BPSK. This causes the error distribution to be: $$n\sim\mathcal{N}(0,N_0)$$ The error probability is thus: $$P(err|s_1)=P(n_3>\sqrt{\varepsilon_b})=Q\left(\sqrt{\frac{\varepsilon_b}{N_0}}\right)$$ The probabilities are equiprobable, being $P(err|s_1)=P(err|s_2)$, so the overall bit error is: $$P_b=Q\left(\sqrt{\frac{\varepsilon_b}{N_0}}\right)$$ For the same energy, binary PAM has better error performance than binary orthogonal signalling. Alternatively, binary PAM can have the same error probability with half the energy. This is because for the same energy, the distance between constellation points is greater for 2-PAM.