Receivers & Detection

The Receiver Problem

• Transmitter knows; Receiver must guess
• r(t) = s_i(t) + n(t)
• Receiver = Estimation + Detection + Decision
• Demodulation extracts, Detection decides

Threshold Detection

• Convert continuous metric to discrete symbol
• If z > γ, decide 1. If z < γ, decide 0.
• Threshold (γ) separates competing hypotheses
• Optimum γ depends on noise & prior probabilities

Probability of Error (P_e)

• Error occurs in the overlapping tails of PDFs
• False Alarm: H₀ true, but decide H₁
• Missed Detection: H₁ true, but decide H₀
• Total P_e relies on noise variance & spacing

ML vs MAP Detection

• Maximum Likelihood (ML): Best matches observation
• Maximum A Posteriori (MAP): Uses Bayes' rule
• MAP incorporates prior probability of symbols
• If priors are equal, MAP reduces to ML

Matched Filter

• Optimized for a specific known waveform
• Maximizes output SNR exactly at sampling instant
• Impulse response is time-reversed known signal
• Mathematically equivalent to a Correlator

Symbol Timing Recovery

• Receiver must know exactly WHEN to sample
• Poor timing → noise sensitivity & severe ISI
• Clock recovery extracts reference from signal
• Methods: Early-late gate, Gardner detector

Carrier Recovery

• Required for coherent PSK/QAM reception
• f_LO ≠ f_carrier → Constellation rotates infinitely
• PLL estimates phase/frequency offset
• Corrects signal before constellation decision

Intersymbol Interference

• Bandwidth limits cause pulses to spread in time
• Tails of one symbol corrupt neighboring symbols
• y_k = a_k + (ISI from others) + noise
• Major limit on high-speed digital throughput

Nyquist Zero-ISI Condition

• Design pulses to have zero crossings at T_s
• h(nT) = 0 for all non-zero integers
• Raised-Cosine & Root-Raised-Cosine (RRC) filters
• Roll-off factor (α) trades bandwidth for realizability

Equalization

• Compensates for channel-induced distortion/ISI
• Zero-Forcing: G(f) ≈ 1/H(f) (can enhance noise)
• MMSE: Balances ISI reduction with noise enhancement
• DFE: Uses previously decided symbols to cancel ISI

Eye Height & Noise

• Superposition of many symbol intervals
• Vertical opening = Amplitude/Noise margin
• Larger eye height → greater noise tolerance
• Amplitude distortion vertically compresses the eye

Eye Width & Jitter

• Horizontal opening = Timing margin
• Larger eye width → tolerant to timing errors
• Jitter (timing variation) causes horizontal closure
• Best sampling instant is exactly at eye center

Error Vector Magnitude

• Distance between ideal constellation point and actual
• Cumulative metric of thermal noise, phase noise, ISI
• Small EVM → highly accurate constellation
• High EVM → likely decision boundary crossings

Hard vs Soft Decisions

• Hard Decision: Receiver outputs strict 1 or 0
• Soft Decision: Outputs confidence/reliability
• Log-Likelihood Ratio (LLR): Magnitude = confidence
• Crucial input for advanced Channel Decoders (Realm 12)

Error Rate (BER/SER)

• BER = wrong bits / transmitted bits
• E_b/N_0 ↑ → BER ↓ (The Waterfall Curve)
• Q-Function evaluates Gaussian tail probabilities
• BPSK AWGN: P_b = Q(√(2E_b/N_0)) is the gold standard

16 — The Detection Laboratory

The single link is now perfect. We have taken analog chaos, filtered it, synchronized to it, equalized it, and extracted the original digital truth. But a communication network rarely consists of just one sender and one receiver. We must now ask: How do thousands of these independent streams share the same physical cable or airspace without destroying each other? You are now prepared for Realm 10: Multiplexing.