Optimum Receiver for M-ary Signals

• A digital communication system transmits digital information using an M-ary signaling scheme.
• The bit stream gets divided into k-bit symbols, resulting in 2^k = M possible symbols.
• These symbols are converted to corresponding signals from the set {sm(t), m = 1, 2, ..., M} and transmitted at each signaling interval T.
• The received signal in an interval is given by r(t) = sm(t) + n(t), where 0 ≤ t ≤ T.
• n(t) represents the sample function of an Additive White Gaussian Noise (AWGN) process with a power spectral density of Sn(f) = N₀/2 watt/Hz.
• The optimum receiver's purpose is to minimize the probability of error

Optimum Receiver Sub-blocks

• Signal Demodulator
• Detector

Signal Demodulator

• Converts the received signal r(t) into an N-dimensional vector r = (r₁, r₂, r₃, r₄, ..., rN) in each signaling interval, with N being the dimension of the transmit signal space.
• ri, where i = 1, 2, ..., N, is the coefficient corresponding to the i-th basis function.
• r = r₁ψ₁(t) + r₂ψ₂(t) + r₃ψ₃(t) + ... + rNψN(t)

Detector

• Decides which of the M possible signal waveforms was transmitted based on observing the received signal vector r in each signaling interval.

Types of Signal Demodulators

• Correlation type demodulator
• Matched filter type demodulator

Correlation-Type Demodulator

• Multiplies the received signal with the carrier (orthonormal basis function), then integrates (using an LPF), and samples the result at every signaling interval.
• For N-dimensional signal demodulation, the signal passes through a parallel bank of N correlators.
• The k-th branch of the correlator multiplies with the k-th basis function ψk(t) integrating the output.
• Integrator output gets sampled, resulting in rk at every sampling interval.
• Mathematically, ∫₀^T r(t)ψk(t)dt = ∫₀^T [sm(t) + n(t)]ψk(t)dt, where k = 1, 2, ..., N and rk = smk + nk.
• Here, smk = ∫₀^T smk(t)ψk(t)dt and nk = ∫₀^T n(t)ψk(t)dt, representing the noise component along the k-th function.
• The signal can be synthesized as r(t) = Σ(k=1 to N) smkψk(t) + Σ(k=1 to N) nkψk(t) = Σ(k=1 to N) rkψk(t).

Matched Filter Type Demodulator

• Passes the received signal r(t) through a parallel bank of N linear filters.
• The impulse response of the filter in the k-th branch is given by hk(t) = ψk(T - t), where 0 ≤ t ≤ T, and k = 1, 2, ..., N.
• N represents the dimension of the signal space, and ψk(t) is the k-th basis function.
• The k-th filter output is: yk(t) = r(t) * hk(t) = ∫₀^t r(τ)hk(t - τ) dτ = ∫₀^t r(τ)ψk(T - t + τ) dτ.
• Sampled at t = T, yk(t) = ∫₀^t r(τ)ψk(τ) dτ.
• For a signal s(t) confined to a time interval of 0≤t≤T, the matched filter has an impulse response of h(t) = s(T - t).

Optimum Detector

• Receives a vector r = (r₁, r₂, r₃, r₄, ..., rN) from the demodulator output, which is the sum of the transmitted signal sm and the noise vector n.
• The sm vector represents a signal at a point on an N-dimensional signal space.
• The n vector is an N-dimensional random vector following an independent Gaussian (Normal) distribution, with a zero mean and a variance of N₀/2.
• The noise vector n possessing spherical symmetry in N-dimensional space.
• When symbol sm gets transmitted, the received signal vector r gets represented by a spherical cloud centered at sm.
• The value N₀/2 defines the density of the noise cloud.
• Task makes a decision on the signal sm from observation r in each interval with the least probability of error.

Decision Criterion

• Criterion formed is centered on the posterior probability, defined by the probability of transmitting sm, given that r is observed.
• Criterion picks the signal having maximum posterior probability from the potential probabilities {P(sm/r), m = 1, 2, ..., M}.
• The estimate of the transmitted signal is: ŝ = arg max P(sm/r).
• This approach is called the maximum a posterior (MAP) criterion.
• In the special instance where all M symbols are equiprobable (i.e., P(sm) = 1/M, m = 1, 2, ..., m), the condition picks the signal having minimum Euclidean distance from the observation point.
• ŝ = arg max Σ(k=1 to N) ||rk - smk||².
• The decision criterion is called the maximum likelihood (ML) criterion.

Probability of Error

• Probability represents the likelihood of the detector making a wrong decision.
• For a binary signal, the detector identifies s₁ when s₂ is transmitted, or vice versa.
• Bit error rate (BER) refers to the probability of errors based on the number of bit errors that occurred relative to the total bits sent.

Binary Antipodal Signal

• Binary antipodal signals are represented with a single-dimensional vector space
• s₁ = √Eb and s₂ = -√Eb, with both signals being equally likely.
• The received signal is r = {s₁ + n = √Eb + n} or {s₂ + n = -√Eb + n}.
• n is an additive noise component that follows a Gaussian distribution with zero mean and variance N₀/2.
• The conditional PDF of the received signal r when s₁ is transmitted is: f(r/s₁) = (1/√(πN₀)) * exp(-(r - √Eb)² / N₀). Conversely, for s₂, f(r/s₂) = (1/√(πN₀)) * exp(-(r + √Eb)² / N₀).
• For equiprobable signals, the formula to calculate the threshold is TH = (√Eb - √Eb) / 2 = 0.
• The detector opts to choose s₁ when r > 0; otherwise, it selects s₂. The likelihood of error if s₁ is the signal sent: P(e/s₁) = ∫(-∞ to 0) f(r/s₁)dr. Where P(e/s₁) = ∫(-∞ to 0) (1/√(πN₀)) * exp(-(r - √Eb)² / N₀) dr.
• The probability of error hinges on the ratio between bit energy (Eb) and noise PSD (N₀), defining the signal-to-noise ratio (SNR).
• Given that the Q-function has an inverse relation with SNR, boosting SNR levels reduces error probability.
• The Euclidean distance between two signals on a vector space: d = 2√Eb.
• Error calculation formula: Pe = Q(√(d² / 2N₀)).

Binary Orthogonal Signal

• Binary orthogonal signal: Two signal points are represented with two orthonormal basis functions.
• The Euclidean distance between two signal points on vector space: d = √2Eb.
• Pe = Q(√(Eb / N₀)).
• Q(0) = 0.5, i.e., the maximum bit-error-rate is 0.5.