Digital Communication – Random Processes to M-ary PSK

Random Processes & Noise

• Random Variable (RV) vs Random Process (RP)

  • Random Variable (RV): A function that maps each outcome in the sample space of a random experiment to a real number. For example, the outcome of a single coin flip mapped to 0 or 1.

  • Random Process (RP) $X(t)$: A family of random variables indexed by time $t$ (or sometimes space); for each outcome of the underlying experiment, a complete function of time (a waveform) is generated. Unlike an RV, which gives a single value, an RP describes the evolution of a random phenomenon over time. Each outcome yields a unique sample function or realization.

• Ensemble & Sample Function

  • Ensemble: The collection of all possible sample functions (waveforms) that a random process can produce. It represents all possible 'histories' of the random phenomenon.

  • Sample function: A single specific waveform or realization obtained from the random process for one particular outcome of the underlying experiment. For instance, if you measure noise on a wire over time, one such measurement trace is a sample function.

• Stationarity

  • Strict-sense Stationarity (SSS): A random process is SSS if all its joint cumulative distribution functions (CDFs) remain invariant to any shift in time. This means the statistical properties (like mean, variance, higher-order moments) are constant over time.

  • Wide-sense Stationarity (WSS): A less restrictive form of stationarity. A random process is WSS if and only if:

    1. Its mean $m_x = E[X(t)]$ is constant and independent of time.

    2. Its autocorrelation function $R_x( au) = E[X(t)X(t+ au)]$ depends only on the time difference $\tau$ (lag) and not on the absolute time $t$. WSS is commonly assumed in practical communication systems because it simplifies analysis while still capturing essential statistical behaviors.

  • Cyclostationary Process: A type of non-stationary process whose statistical properties (mean, autocorrelation) vary periodically with time. Often observed in modulated signals.

• Ergodicity (in mean / autocorrelation)

  • A random process is ergodic (e.g., in mean or autocorrelation) if its time averages (calculated over a single, infinitely long sample function) are equal to its ensemble averages (calculated across all sample functions at a fixed time). This property is crucial in practice as it allows us to infer statistical properties of a random process by performing measurements on just a single sufficiently long sample function, rather than requiring an infinitely large ensemble of measurements.

• Moments for WSS RP

  • Mean $m_x = E[X(t)]$: The average value of the random process, which is constant for a WSS process.

  • Autocorrelation $R_x( au) = E[X(t)X(t+ au)]$: Measures the statistical similarity between the process at time $t$ and at time $t+ au$. For a WSS process, this only depends on the time lag $\tau$. It provides information about the power spectral density of the process.

  • Auto-covariance $C<em>x( au) = R</em>x( au) - m<em>x^2$: Similar to autocorrelation but measures the correlation around the mean. For a zero-mean process, $C</em>x( au) = R_x( au)$.

• Einstein–Wiener–Khintchine Theorem
  $S<em>x(f) = \int</em>{-\infty}^{\infty} R<em>x(\tau)e^{-j2\pi f\tau}d\tau$ $R</em>x(\tau) = \int<em>{-\infty}^{\infty} S</em>x(f)e^{j2\pi f\tau}df$
  This theorem states that the Power Spectral Density (PSD) $S<em>x(f)$ and the autocorrelation function $R</em>x(\tau)$ of a WSS random process form a Fourier transform pair. This fundamental relationship allows us to analyze the frequency content of a random process from its time-domain correlation properties, and vice-versa.

• White Gaussian Noise (WGN)

  • Power Spectral Density (PSD): $S<em>n(f) = \frac{N</em>0}{2}$ (flat across all frequencies). This means WGN has equal power per unit bandwidth at every frequency, analogous to white light containing all colors. $N_0$ is the noise power/Hz.

  • Autocorrelation Function: $R<em>n(\tau) = \frac{N</em>0}{2}\delta(\tau)$. The autocorrelation function is an impulse at $\tau = 0$, implying that noise samples at different time instants are uncorrelated (and statistically independent if the noise is Gaussian). The 'Gaussian' part means its amplitude distribution is Gaussian.

  • WGN is a crucial model in communication systems because it simplifies analysis and accurately approximates thermal noise encountered in electronic components.

• Linear filtering of WSS RP
  When a WSS random process $X(t)$ with mean $m<em>x$ and PSD $S</em>x(f)$ passes through a Linear Time-Invariant (LTI) filter with impulse response $h(t)$ and frequency response $H(f)$, the output process $Y(t)$ will also be WSS:

  • Output mean $m<em>y = m</em>x H(0)$. The DC component of the input mean is scaled by the filter's DC gain.

  • Output PSD $S<em>y(f) = |H(f)|^2 S</em>x(f)$. The filter shapes the input PSD, with the output power spectrum determined by the magnitude-squared of the filter's frequency response multiplied by the input PSD.

  • Output power $E[Y^2] = \int<em>{-\infty}^{\infty} |H(f)|^2 S</em>x(f)df$

    • For zero-mean processes, power is equal to variance: $E[Y^2] = \sigma<em>y^2 = R</em>y(0)$. This integral calculates the total power of the output process, representing the area under the output PSD.

Integrate-and-Dump (I&D) Receiver

• An Integrate-and-Dump (I&D) receiver is an optimal filter for detecting rectangular baseband pulses in the presence of AWGN. It works by integrating the received signal over the duration of one bit period, $T_b$, and then