Communication, Sensing, and Multirate Applications

Introduction and context

Signals and systems theory becomes most valuable when it is used to move information, infer physical conditions, and process data at different time or frequency scales. Communication systems transform information into waveforms suitable for transmission through a channel and recover an estimate at the receiver. Sensing systems observe a physical process through transducers and infer unknown quantities from measured signals. Multirate systems deliberately change sampling rates to achieve efficient, flexible, and computationally economical processing.

These three areas are closely connected. A wireless receiver is a communication system, but it also performs sensing through received signal strength, delay, Doppler shift, and angle of arrival. A radar is primarily a sensing system, but it uses communication-like modulation and detection. A software-defined radio depends on multirate filtering and sampling-rate conversion. At postgraduate level, the central task is not merely to identify blocks in a diagram, but to analyze bandwidth, noise, aliasing, estimation error, computational complexity, and implementation constraints together.

Learning objectives

After studying this chapter, the reader should be able to:

1. Relate signal-space, spectral, and system-level descriptions of communication and sensing systems.
2. Analyze sampling, reconstruction, aliasing, and sampling-rate conversion.
3. Explain interpolation, decimation, polyphase structures, and multistage conversion.
4. Derive and interpret basic performance measures such as signal-to-noise ratio, bit error probability, range resolution, and estimation uncertainty.
5. Design an appropriate processing chain for a practical communication or sensing application.
6. Evaluate limitations involving noise, synchronization, finite precision, interference, privacy, and electromagnetic spectrum use.

1. A unified signal-and-system model

A useful abstract model is

where x(t) is the transmitted or physical source signal, 𝓗 denotes the channel or measurement operator, r(t) is the received or measured signal, and n(t) is noise or unmodelled disturbance. In a linear time-invariant approximation, 𝓗 is convolution with an impulse response h(t):

The model is applicable when propagation or sensing dynamics can reasonably be treated as linear and time invariant over the observation interval. In mobile radio, radar, biomedical sensing, and acoustic systems, h(t) may vary with time; then a time-varying or stochastic channel model is more appropriate.

A digital processing chain usually contains four transformations:

• Representation: conversion of information into a signal or sequence.
• Transmission or observation: interaction with a physical channel or environment.
• Inference: detection, demodulation, estimation, or classification.
• Decision and control: production of data, alarms, commands, or feedback.

A system should be evaluated at several levels. The spectral level asks how energy is distributed over frequency. The temporal level asks how delay, duration, and transients behave. The statistical level asks how noise and uncertainty affect decisions. The implementation level asks whether the required processing can be performed with available sampling rate, memory, latency, and power.

2. Communication applications

2.1 Baseband and passband representations

A real passband waveform can often be represented using an equivalent complex baseband signal:

Here s_p(t) is the real passband signal, s(t) is the complex envelope, f_c is carrier frequency in hertz, and Re{·} denotes the real part. This representation assumes that the complex envelope occupies a bandwidth much smaller than the carrier frequency and that the modulation is centered around f_c.

Writing

shows that in-phase I(t) and quadrature Q(t) components jointly describe amplitude and phase. Quadrature modulation is important because it permits two signal dimensions in the same symbol interval. The symbol energy is

where the integral extends over the symbol interval and the signal is expressed in a consistent impedance-normalized form. If the symbol duration is T_s and the average symbol energy is E_s, the symbol rate is R_s = 1/T_s.

2.2 Channel capacity and link quality

For an additive white Gaussian noise channel with bandwidth B and received signal-to-noise ratio S/N, the ideal capacity is

C is in bits per second when B is in hertz and S/N is dimensionless. This is an upper bound for reliable communication under the assumptions of Gaussian noise, ideal coding, sufficiently long codewords, and a channel whose relevant parameters are known or appropriately estimated. It is not the throughput of a particular modem.

The ratio S/N is often expressed in decibels:

where P_s and P_n are signal and noise powers measured over the same bandwidth. A voltage ratio may use 20 log₁₀ only when the impedances are equal.

For a received power P_r, noise spectral density N₀, and bandwidth B, the noise power under a white-noise model is P_n = N₀B. Thus increasing bandwidth can increase capacity but also increases integrated noise. Engineering design is therefore a trade-off among bandwidth, transmit power, coding, latency, and hardware complexity.

2.3 Detection, synchronization, and estimation

A receiver does more than apply a demodulator. It must estimate timing, carrier frequency, carrier phase, channel response, and sometimes the number of active users. A matched filter maximizes the sampled signal-to-noise ratio at a selected time for a known waveform in additive white Gaussian noise. Its impulse response is proportional to a time-reversed, conjugated version of the expected signal.

For binary antipodal signaling in additive white Gaussian noise, the ideal coherent bit-error probability is

where E_b is energy per bit, N₀ is the one-sided noise power spectral density, and Q(x) is the tail probability of a standard normal random variable. This formula assumes perfect synchronization, equally likely bits, coherent detection, and an ideal channel. Fading, phase error, nonlinear distortion, and coding change the result.

Synchronization errors are often more damaging than a modest reduction in nominal SNR. A timing error can introduce intersymbol interference; carrier-frequency offset causes a rotating constellation and, in multicarrier systems, intercarrier interference. Practical receivers therefore use preambles, pilots, feedback loops, training sequences, or blind estimation.

3. Sensing and inference from signals

3.1 Measurement model

A sensing system estimates a parameter θ from observations. A simple model is

where y[k] is the kth sample, A is an amplitude or gain, s is a known or modelled waveform, τ is delay, θ represents unknown parameters, T_s is the sampling interval, and w[k] is measurement noise. The model is useful for radar, sonar, localization, vibration analysis, and instrumentation, although real systems may require multipath, clutter, nonlinear sensors, and time-varying gain.

The sensing objective may be detection, estimation, localization, tracking, or classification. Detection decides between hypotheses, such as target present or absent. Estimation produces a numerical value, such as range or velocity. Localization estimates position. Tracking estimates a state over time using a dynamical model.

3.2 Radar and sonar principles

For a monostatic radar, if a transmitted pulse returns after delay τ, target range R is

where c is propagation speed and the factor 2 accounts for the outward and return paths. For sonar, c is replaced by the speed of sound in the medium. The formula assumes a direct path and negligible processing delay.

For a pulse of duration T_p, approximate range resolution is

for an uncompressed pulse. Shorter pulses improve resolution but contain less energy. Pulse compression resolves this conflict by transmitting a long coded or frequency-swept waveform and applying a matched filter. If the effective compressed bandwidth is B, the approximate range resolution becomes

provided the waveform and receiver support that bandwidth and sidelobes are controlled.

A moving target produces Doppler frequency shift. For a narrowband monostatic radar, the approximate Doppler shift is

where v is radial velocity and λ is wavelength. The sign depends on the convention for approaching and receding motion. This expression assumes nonrelativistic motion and a single dominant radial component.

The central sensing trade-off is that resolution, ambiguity, sensitivity, update rate, and computational cost cannot generally be optimized independently. Wider bandwidth improves delay resolution; longer coherent integration improves detectability but may reduce tolerance to target motion; larger apertures improve angular resolution but increase hardware cost.

3.3 Statistical decision measures

A detector can be characterized by probability of detection P_D and probability of false alarm P_FA. Increasing a decision threshold often decreases P_FA but also decreases P_D. A receiver operating characteristic (ROC) curve displays this trade-off without selecting one threshold as universally correct.

The Neyman–Pearson approach fixes an acceptable false-alarm probability and chooses a detector that maximizes detection probability under that constraint. It is appropriate when false alarms have a known operational cost. Bayesian detection instead incorporates prior probabilities and explicit costs of incorrect decisions. Neither framework eliminates the need for a credible signal and noise model.

For repeated noisy measurements of a constant quantity, averaging can reduce random noise. If independent measurements have variance σ², their arithmetic mean has variance σ²/N, where N is the number of measurements. This improvement does not hold for correlated noise, drift, interference, or systematic calibration error.

4. Sampling and reconstruction

4.1 Sampling theorem

If a continuous-time signal is strictly bandlimited to frequencies |f| ≤ B, ideal uniform sampling permits perfect reconstruction when

where f_s is the sampling frequency. The strict inequality avoids ambiguity at the band edge; in practical systems a margin is required. Reconstruction uses a low-pass interpolation filter. The theorem does not say that every signal can be sampled safely at twice its highest informal frequency: the signal must be bandlimited or filtered before sampling.

Sampling creates spectral replicas at integer multiples of f_s. If replicas overlap, aliasing occurs and cannot be removed after sampling. An analog anti-alias filter therefore precedes an ADC. Its transition band depends on the signal band, sampling rate, allowable attenuation, and converter resolution.

Quantization introduces amplitude error. For an ideal uniform quantizer with step size ∆ and sufficiently busy input, the quantization-noise power is approximately

This approximation assumes the quantization error is uncorrelated with the signal and uniformly distributed over [−∆/2, ∆/2]. It is unreliable for very small deterministic signals, overload, or correlated inputs. For an ideal N-bit ADC with a full-scale sinusoid, the commonly used approximate SNR is

The result assumes an ideal converter, a full-scale sinusoid, and quantization noise spread over the Nyquist band.

4.2 Sampling-bandpass signals

A bandpass signal need not always be sampled at more than twice its highest carrier frequency. With deliberate bandpass sampling, spectral replicas can be arranged not to overlap, allowing a rate related to occupied bandwidth rather than carrier frequency. However, the permitted sampling-rate intervals depend on the band edges, and analog filtering must prevent unwanted bands from folding into the same aliased band. Bandpass sampling is therefore a frequency-planning technique, not a violation of the sampling theorem.

5. Multirate signal processing

Multirate processing changes the sampling rate while preserving the desired information. The fundamental operations are decimation, interpolation, and rational sample-rate conversion.

5.1 Decimation

Decimation by an integer M retains every Mth sample:

Direct decimation is safe only when x[n] has first been low-pass filtered so that its spectrum occupies no more than approximately π/M radians per sample. The anti-alias filter is essential because downsampling compresses the discrete-time spectrum and can overlap spectral replicas.

In a practical decimator, filtering precedes sample deletion:

where h[n] is the anti-alias impulse response. The output rate is f_out = f_in/M.

5.2 Interpolation

Interpolation by L consists conceptually of inserting L−1 zeros between samples and low-pass filtering. If the expanded sequence is u[n],

and the output is y[n] = u[n] * h[n]. The interpolation filter removes spectral images caused by zero insertion. The output sampling rate is Lf_in.

The ideal low-pass cutoff must be chosen consistently with the new sampling frequency and desired signal bandwidth. In normalized discrete-time frequency, the cutoff for the interpolation filter is typically at or below π/L before frequency normalization to the output rate.

5.3 Rational conversion and polyphase implementation

To convert from f_in to f_out with ratio L/M, interpolate by L and then decimate by M. The output rate is

where L and M are positive integers, preferably reduced to lowest terms. Filtering should be placed between the two conceptual operations. In implementation, a polyphase filter decomposes h[n] into phases and computes only output samples that are actually needed. This avoids explicitly generating zero-valued samples and can reduce computation approximately in proportion to the conversion ratio.

For a prototype filter h[n] and interpolation factor L, one polyphase decomposition is

where e_r[k] is the rth phase. Equivalent decompositions can be used for decimators and rational converters. The important principle is that filtering and rate change are reorganized without changing the ideal input-output operation.

5.4 Multistage conversion

A large conversion ratio is often factored into several smaller stages. For example, conversion by 1/12 can be implemented as 1/3 followed by 1/4, or in another factorization. Multistage design can reduce the number of filter multiplications because each stage operates at a different rate and can use a transition band appropriate to that rate. The design must still account for cumulative passband droop, stopband attenuation, coefficient quantization, and stage-by-stage alias protection.

Half-band filters are particularly efficient for conversion by two because, ideally, approximately half their alternating coefficients are zero. Cascaded-integrator-comb (CIC) filters are multiplier-free structures useful for large integer rate changes, especially in hardware. Their drawbacks include passband droop and limited control of the transition band; a compensating FIR filter is often required.

6. Worked example: a software-defined receiver

Consider a receiver whose ADC samples an intermediate-frequency signal at 96 kHz. The desired channel has bandwidth 8 kHz, and the digital architecture must produce complex baseband samples at 24 kHz.

First, the rate conversion ratio is

Thus a decimator by M = 4 is sufficient. Before decimation, a digital low-pass filter must limit the signal to a bandwidth below approximately 12 kHz, the Nyquist frequency of the 24-kHz output. In practice, the passband might end at 8 kHz and the stopband might begin below 12 kHz, with the exact transition selected from allowable distortion and computational cost.

If the signal is still centered at an intermediate frequency, digital mixing is required before or within the decimation chain. A complex oscillator e^(−j2πf_0n/f_s) translates the channel by f_0. The mixer must be followed by suitable filtering, because multiplication creates sum and difference frequency components. A practical chain is therefore:

1. ADC and analog anti-alias filtering.
2. Digital frequency translation to complex baseband.
3. Low-pass channel filtering.
4. Decimation by four using a polyphase FIR filter.
5. Timing, carrier, and symbol recovery.
6. Demodulation and decoding.

The example illustrates an important design principle: rate conversion is not merely deletion or insertion of samples. It is a spectral operation that must be integrated with channel selection and synchronization.

7. Applied case: illustrative industrial condition monitoring

Background

Consider an illustrative factory monitoring system for a high-speed motor. An accelerometer produces vibration data at 48 kHz. The maintenance team wants a low-power wireless node that detects bearing faults and transmits diagnostic features rather than the entire waveform.

Problem

Bearing defects may generate impulsive components and characteristic spectral energy. Directly transmitting 48-kHz data consumes energy and wireless bandwidth. However, excessive downsampling could remove fault information or cause aliasing. The system must also distinguish genuine faults from speed changes and sensor noise.

Analysis

The design team first identifies the frequency bands associated with normal rotation, harmonics, and bearing resonance. An analog anti-alias filter protects the ADC. A digital bandpass filter isolates the diagnostic band. If the diagnostic band is below 4 kHz, the data can be decimated from 48 kHz to 10 kHz through a rational converter with ratio 5/24, implemented in stages. Features include band energy, crest factor, spectral kurtosis, and envelope-spectrum peaks.

For a sequence x[k] of N samples, band energy can be estimated as

where x_B[k] is the output of the selected bandpass filter. The feature is meaningful only if sensor calibration, window duration, sampling rate, and normalization are documented. Spectral features should be compared against a baseline at comparable motor speed and load.

Approach and solution

The node uses a high-resolution ADC, a multistage decimator, windowed spectral analysis, and a lightweight classifier. It stores raw snippets around alarms so that engineers can audit decisions. A rate-adaptive stage handles changes in the available wireless link, while local processing retains a fixed diagnostic bandwidth.

Outcome

The architecture reduces transmitted data and radio-on time while retaining the frequency components selected by domain experts. It also makes the diagnostic decision explainable: an alarm can be associated with a measured band-energy increase and a change in envelope modulation rather than being an unexplained label.

Learning points

1. Sampling-rate reduction is valid only after defining the information-bearing bandwidth.
2. Feature extraction must account for operating conditions, not only nominal frequency.
3. A classifier cannot correct information destroyed by aliasing.
4. Retaining audit snippets improves maintenance, validation, and safety.
5. Multirate processing can reduce energy without sacrificing useful information when its filters are designed from the sensing objective.

8. Design limitations and professional practice

Noise, interference, and nonstationarity

Communication and sensing channels are rarely white, stationary, or independent. Adjacent-channel interference, impulsive noise, multipath, clutter, oscillator phase noise, and electromagnetic coupling may dominate thermal noise. A design based only on average SNR can therefore be misleading. Measurements should include worst-case interference, dynamic range, crest factor, and time variation.

Finite precision and implementation

FIR and IIR coefficients are quantized in hardware. Fixed-point systems require scaling, overflow analysis, saturation policy, and stability checks. Narrow transition bands require longer filters, increasing latency and computation. IIR filters may be efficient but need careful pole quantization and state management. In a multistage converter, word length and noise growth must be analyzed at each stage.

Latency and reliability

A communication receiver may prioritize throughput, whereas a control or protection system may prioritize deterministic latency. A sensing system may tolerate delayed batch analysis but not delayed fault protection. Buffering, block transforms, error correction, retransmission, and adaptive filtering all affect latency and reliability. Requirements must therefore be stated quantitatively rather than described as simply “real time.”

Spectrum, privacy, and governance

Radio systems must comply with national spectrum allocations, emission masks, power limits, and coexistence rules. Sensing systems can reveal location, activity, health, or industrial secrets. Data minimization, access control, encryption, retention limits, and informed governance are engineering requirements, not optional additions. Machine-learning decisions should be validated across operating conditions and monitored for distribution shift. In safety-critical applications, human override and traceable logs are essential.

9. Contemporary developments

Software-defined radio increasingly uses wideband converters, direct RF sampling, digital channelizers, and programmable FPGA or ASIC pipelines. Channelizers use filter banks to divide a wideband input into many subchannels; polyphase filter banks provide an efficient implementation by combining filtering with a discrete Fourier transform.

Joint communication and sensing uses shared waveforms and hardware to transmit information while estimating range, velocity, or angle. The approach can improve spectrum utilization but introduces waveform-design conflicts: communication symbols may increase sensing sidelobes, while sensing-oriented waveforms may reduce data efficiency.

Distributed sensing and cooperative localization combine observations from spatially separated devices. Their performance depends on synchronization, calibration, data association, and communication overhead. Edge AI can reduce latency and bandwidth by extracting features locally, but model uncertainty, adversarial inputs, explainability, and retraining must be addressed.

Best practice is to begin with an explicit signal model and measurable requirements. Then define bandwidth, dynamic range, sampling rates, allowable aliasing, latency, probability of detection, false-alarm constraints, and computational budget. Simulation should be followed by hardware-in-the-loop testing and measurements under nonideal conditions.

Chapter Summary

Communication systems represent information, transmit it through a physical channel, and infer the intended message under noise and distortion. Complex envelopes, SNR, channel capacity, matched filtering, synchronization, and error probability provide the principal analytical tools. Sensing systems use measured waveforms to detect, estimate, localize, or classify physical phenomena; delay and Doppler connect signal measurements to range and velocity, while ROC analysis expresses detection trade-offs.

Sampling theory explains when continuous-time information can be represented digitally. Anti-alias filtering is indispensable before decimation, and reconstruction or anti-imaging filtering is required after interpolation. Rational sample-rate conversion uses interpolation by L and decimation by M, while polyphase, multistage, half-band, and CIC structures improve implementation efficiency. Practical systems must additionally address finite precision, time variation, synchronization, latency, spectrum regulation, privacy, and explainability. The most reliable designs begin with the information-bearing bandwidth and operational objective, not with a convenient sampling rate or algorithm.

Key Terms

Complex envelope: Equivalent complex baseband representation of a passband waveform.

Quadrature components: In-phase and quadrature signal components forming two orthogonal dimensions.

Channel capacity: Theoretical maximum reliable information rate under specified channel assumptions.

Matched filter: Filter optimized for peak output SNR for a known waveform in white noise.

Synchronization: Estimation and correction of timing, frequency, and phase references.

Sensing: Inferring physical conditions or parameters from measured signals.

Range resolution: Ability to distinguish targets separated in propagation distance.

Doppler shift: Frequency change caused by relative radial motion.

Probability of detection: Probability that a detector correctly declares a target or signal present.

False-alarm probability: Probability of declaring a target or signal when it is absent.

Aliasing: Irrecoverable spectral overlap caused by inadequate sampling or downsampling.

Anti-alias filter: Filter that limits input bandwidth before sampling or decimation.

Decimation: Reduction of sampling rate, normally preceded by low-pass filtering.

Interpolation: Increase of sampling rate using zero insertion and anti-imaging filtering.

Polyphase filter: Decomposed filter structure that efficiently combines filtering and rate conversion.

Rational sample-rate conversion: Conversion by a ratio L/M using interpolation and decimation.

Quantization noise: Error introduced by representing amplitudes with finite resolution.

ROC curve: Plot of detection probability against false-alarm probability as threshold varies.

Channelizer: Filter-bank system that separates a wideband signal into subchannels.

Joint communication and sensing: Use of shared signals and infrastructure for data transfer and environmental inference.

Review Questions

Short Answer Questions

1. Why must an anti-alias filter precede a decimator?
2. Define the complex envelope of a passband signal.
3. State two assumptions behind the Shannon capacity formula.
4. What physical quantity is estimated from Doppler shift in radar?
5. Distinguish probability of detection from probability of false alarm.

Long Answer Questions

1. Explain the complete signal-processing chain of a software-defined radio, including sampling, frequency translation, filtering, synchronization, and decoding.
2. Derive the conceptual operations required for rational sample-rate conversion by L/M and explain why polyphase implementation is efficient.
3. Discuss the resolution, sensitivity, and ambiguity trade-offs in pulsed radar and pulse-compression radar.
4. Explain how quantization, finite coefficient precision, and overflow influence a fixed-point multirate implementation.
5. Compare Bayesian and Neyman–Pearson detection approaches and identify situations in which each is appropriate.

Analytical / Application Questions

1. A sensor is sampled at 40 kHz and its useful band is 0–6 kHz. Propose a safe decimation factor and specify the essential filter constraints.
2. Design a conceptual vibration-monitoring chain that converts 48-kHz accelerometer data to a lower rate while preserving a 0–4-kHz diagnostic band. Justify each stage.
3. A radar operates at wavelength 0.03 m and observes a target with radial speed 15 m/s. Using the narrowband monostatic approximation, calculate the Doppler-shift magnitude and discuss assumptions that could invalidate the result.

Multiple Choice Questions with Answers

Question 1

For an ideal AWGN channel with bandwidth B and signal-to-noise ratio S/N, which expression gives the theoretical capacity?

A. C = B(S/N)

B. C = B log₂(1 + S/N)

C. C = 2B log₁₀(S/N)

D. C = S/N − B

Correct Answer: B. C = B log₂(1 + S/N)

Question 2

What is the primary purpose of an anti-alias filter before decimation?

A. To increase the sampling rate

B. To create carrier synchronization

C. To limit bandwidth before spectral replicas overlap

D. To quantize amplitudes uniformly

Correct Answer: C. To limit bandwidth before spectral replicas overlap

Question 3

A radar measures a round-trip delay τ. Under the direct-path monostatic assumption, range is

A. cτ

B. 2cτ

C. τ/(2c)

D. cτ/2

Correct Answer: D. cτ/2

Question 4

A sample-rate converter changes 96 kHz to 24 kHz. Which integer decimation factor is required?

A. 2

B. 3

C. 4

D. 6

Correct Answer: C. 4

Question 5

For ideal binary antipodal coherent signaling in AWGN, the bit-error probability is

A. Q(√(2E_b/N₀))

B. Q(E_b/N₀)

C. 1 − Q(√(2E_b/N₀))

D. exp(−N₀/E_b)

Correct Answer: A. Q(√(2E_b/N₀))

Question 6

Which operation increases the sampling rate by inserting samples and then applying a low-pass filter?

A. Decimation

B. Interpolation

C. Detection

D. Demodulation

Correct Answer: B. Interpolation

Question 7

For a monostatic radar with wavelength λ, the narrowband Doppler magnitude for radial speed v is approximately

A. v/λ

B. λ/(2v)

C. 2v/λ

D. 2λv

Correct Answer: C. 2v/λ

Question 8

What is a major advantage of a polyphase rate converter?

A. It eliminates the need for filtering

B. It computes many unnecessary zero-valued samples

C. It combines filtering and rate change efficiently

D. It guarantees zero quantization noise

Correct Answer: C. It combines filtering and rate change efficiently

Question 9

The approximate quantization-noise power of an ideal uniform quantizer with step size ∆ is

A. ∆²/12

B. 12∆²

C. ∆/12

D. 1/∆²

Correct Answer: A. ∆²/12

Question 10

On a detector ROC curve, changing the decision threshold generally changes

A. Only the sampling frequency

B. The trade-off between detection probability and false-alarm probability

C. The propagation speed

D. The ADC number of bits

Correct Answer: B. The trade-off between detection probability and false-alarm probability