Communication System

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This section introduces the basic building blocks common to analog and digital communication systems, and the fundamental role of signal and noise in determining system performance.

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Basic Building Blocks Every communication system shares the same essential chain:

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Information source → Transmitter → Channel → Receiver → Destination, with noise entering primarily at the channel.

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The transmitter converts the message into a form suitable for the channel (typically involving modulation, and for digital systems, source/channel coding); the receiver reverses this process to recover the message.

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System Type Key Characteristic Analog communication system Message signal is continuous-valued; modulated directly onto a carrier (e.g.

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AM/FM radio); simpler but more susceptible to noise accumulation Digital communication system Message converted to discrete symbols/bits before transmission; more complex but offers noise immunity, regeneration, encryption, and error control Signal & Noise Noise is any unwanted random disturbance that corrupts the transmitted signal, arising from the channel (external interference) or the receiver's own components (internal, e.g. thermal noise).

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Signal-to-Noise Ratio (SNR) is the primary figure of merit for a communication link — the ratio of signal power to noise power, usually expressed in dB, directly affecting the achievable data rate/quality.

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Digital systems can use regeneration (detecting and re-transmitting clean pulses at intermediate repeater points) to prevent noise from accumulating over long distances, an advantage analog systems lack.

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This section covers the classification of signals/systems as low pass or band pass, the concept of system bandwidth, the requirements for distortionless transmission, and the Hilbert transform and its applications.

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Low Pass & Band Pass Signals/Systems Type Description Low pass signal/system Significant frequency content concentrated around DC (0 Hz) up to some cutoff; e.g. baseband audio/video signals Band pass signal/system Significant frequency content concentrated around some non-zero center frequency, within a band; e.g. a modulated RF/carrier signal Bandwidth of a system: the range of frequencies over which the system passes signal content effectively (e.g. between half-power points for a filter/amplifier), determining the maximum data rate/information content it can support.

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Distortionless Transmission A system provides distortionless transmission if its output is a scaled and time-delayed (but otherwise unchanged) replica of the input: y(t) = K·x(t−t₀).

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This requires the system's magnitude response to be constant (flat) across the signal's bandwidth, and its phase response to be linear with frequency (constant group delay) — any deviation introduces amplitude or phase (delay) distortion.

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The Hilbert Transform The Hilbert transform of a signal produces a 90° phase-shifted version of that signal (shifting each positive-frequency component by −90° and each negative-frequency component by +90°), without changing its magnitude spectrum.

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Applications: constructing the analytic signal (used to define instantaneous amplitude/phase/frequency), generating single-sideband (SSB) modulated signals, and envelope detection.

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This section covers time- and frequency-domain representations of AM, FM, and PM signals, their modulation/demodulation and bandwidth requirements, digital modulation (ASK/FSK/PSK), and M-ary data communication.

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AM, FM & PM — Time & Frequency Domain Scheme Time Domain Expression / Key Feature Amplitude Modulation (AM) s(t) = [A_c + m(t)]cos(ω_ct); carrier amplitude varies with message; spectrum = carrier + two sidebands Frequency Modulation (FM) s(t) = A_ccos(ω_ct + k_f∫m(τ)dτ); carrier frequency varies with message; spectrum has infinite Bessel-function sidebands Phase Modulation (PM) s(t) = A_ccos(ω_ct + kₚₘ(t)); carrier phase varies with message; spectrum similarly Bessel-function based Types of AM signals: conventional (full) AM, DSB-SC (suppressed carrier), SSB-SC (single sideband), VSB (vestigial sideband) — trading off power/bandwidth efficiency against receiver complexity.

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Types of FM signals:

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Narrowband FM (β«1, bandwidth ≈ 2fₘ) and Wideband FM (β>1, bandwidth via Carson's rule ≈ 2(Δf+fₘ)).

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Modulation, Demodulation & Bandwidth AM demodulation: envelope detection (simple, conventional AM) or synchronous/coherent detection (required for DSB-SC/SSB-SC).

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FM demodulation: frequency discriminators or phase-locked loop (PLL) based detectors.

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AM bandwidth ≈ 2fₘ (conventional/DSB-SC) or fₘ (SSB-SC);

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FM bandwidth (Carson's rule) ≈ 2fₘ(β+1), generally much larger than AM for the same message bandwidth.

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Digital Modulation & M-ary Systems Scheme Description ASK (Amplitude Shift Keying) Digital data represented by switching carrier amplitude between discrete levels FSK (Frequency Shift Keying) Digital data represented by switching carrier frequency between discrete values PSK (Phase Shift Keying) Digital data represented by switching carrier phase between discrete values; generally the most noise-resistant of the three M-ary data communication Uses M > 2 distinct symbol states to encode log₂M bits per symbol, improving bandwidth efficiency at the cost of requiring higher SNR (e.g.

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This section covers the analog-to-digital conversion process, source coding, pulse modulation techniques including PCM, quantization types and noise, the Shannon-Hartley channel capacity theorem, and multiplexing.

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Analog-to-Digital Communication Process The standard process:

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Sampling (at ≥ Nyquist rate) → Quantization (mapping to discrete amplitude levels) → Encoding (binary representation) → transmission → decoding/reconstruction at the receiver.

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Source coding removes redundancy from the digitized data to reduce the number of bits required for transmission (e.g.

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Huffman coding), improving efficiency without losing essential information.

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Pulse Modulation & PCM Technique Description Pulse Amplitude Modulation (PAM) Pulse amplitude varies in proportion to the sampled message value Pulse Code Modulation (PCM) Sampled and quantized values are encoded as binary codewords; the standard digital representation of an analog signal Pulse Position/Width Modulation Information carried by the pulse's position or width instead of amplitude (less common but noise-resistant in specific ways) Quantization & Quantization Noise Type Description Uniform quantization Equal step sizes across the entire signal range; simple, but poor SNR for low-amplitude signals Non-uniform quantization Smaller step sizes for low-amplitude signals, larger for high-amplitude (achieved via companding, μ-law/A-law); improves SNR for weak signals Quantization noise/error The difference between the actual sample and its quantized value; for uniform quantization, maximum error = ±(step size)/2 Channel Capacity & Multiplexing Shannon-Hartley theorem:

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C = Blog₂(1+S/N) — gives the maximum error-free channel capacity C (bits/s) for bandwidth B and signal-to-noise ratio S/N.

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Multiplexing combines multiple signals onto one shared channel:

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FDM (frequency division), TDM (time division), and (for optical links) WDM (wavelength division).

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This section introduces information theory fundamentals, measures of information, line coding schemes, pulse shaping to control intersymbol interference, and error control coding techniques.

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Information Theory Fundamentals Information is a measure of the uncertainty resolved by receiving a message; a message that is more surprising (less probable) carries more information.

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Measure of information (self-information):

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I = log₂(1/p) bits, for an event of probability p — less probable events carry more information.

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Entropy: the average information per symbol from a source, H = Σp_ilog₂(1/p_i), representing the theoretical minimum average number of bits needed to represent the source's output.

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Line Codes Line Code Description Unipolar NRZ 1 = positive pulse, 0 = no pulse; simple but has DC component and no self-clocking Polar NRZ 1 = positive pulse, 0 = negative pulse; no DC component issue as severe, but still lacks self- clocking for long runs of same bit Bipolar (AMI) Alternates the polarity of successive 1s (0 = no pulse); no DC component, provides some error detection Manchester code Encodes each bit as a transition at the bit-interval midpoint; self-clocking (guarantees a transition every bit), but requires double the bandwidth of NRZ Pulse Shaping Pulse shaping controls a transmitted pulse's spectral/time-domain shape to limit bandwidth while controlling Inter-Symbol Interference (ISI) — the overlapping of adjacent symbol pulses that can cause detection errors.

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The raised-cosine pulse is a widely used pulse shape satisfying the Nyquist criterion for zero ISI at the correct sampling instants, while offering a tunable trade-off (via its roll-off factor) between bandwidth and time-domain pulse decay.

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Error Control Coding Error control coding adds redundancy to detect and/or correct transmission errors: block codes (e.g.

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Hamming code, cyclic codes/CRC) and convolutional codes (decoded via the Viterbi algorithm) are the two major families.

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This final section of the chapter covers random signals and processes, key noise models (white noise, thermal noise, bandlimited white noise), and the power spectral density and autocorrelation function used to characterize them.

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Random Signals & Processes A random signal (stochastic process) cannot be predicted exactly in advance; it is described statistically by properties such as mean, variance, and correlation, rather than by a deterministic formula.

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A process is stationary if its statistical properties do not change over time; it is wide-sense stationary (WSS) if its mean is constant and its autocorrelation depends only on the time difference between samples, not on absolute time.

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Ergodicity: a process is ergodic if its time averages (computed from a single sample function) equal its ensemble (statistical) averages — a convenient property allowing practical measurement of statistics from a single observed record.

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Noise Models Noise Type Description White noise An idealized noise model with constant (flat) power spectral density across all frequencies; theoretically infinite total power, used as a convenient analytical approximation Thermal noise Physical noise arising from random electron motion due to temperature; well-approximated as white noise over practical bandwidths, with power spectral density proportional to temperature Bandlimited white noise White noise passed through an ideal bandpass/lowpass filter, giving it finite (realistic) total power while retaining a flat spectrum within the passband PSD & Autocorrelation Function Power Spectral Density (PSD): describes how a random signal's average power is distributed across frequency; for white noise, the PSD is a constant, S(f) = N₀/2 across all frequencies.

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Autocorrelation function (AC function): measures how similar a random process is to a time-shifted version of itself, R(τ) = E[X(t)X(t+τ)]; for a WSS process, it depends only on the lag τ.

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The Wiener-Khinchin theorem relates PSD and autocorrelation: the PSD is the Fourier transform of the autocorrelation function.

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For ideal white noise, the autocorrelation is an impulse at τ=0 (R(τ) = (N₀/2)δ(τ)), reflecting that samples are uncorrelated for any nonzero time separation.