Electronics Guide

Signal Classifications

Signals can be classified in several ways:

• Continuous-time vs. discrete-time: Continuous signals exist for all time values; discrete signals exist only at specific instants.
• Analog vs. digital: Analog signals have continuous amplitude; digital signals have quantized amplitude levels.
• Deterministic vs. random: Deterministic signals can be described mathematically; random signals require statistical description.
• Periodic vs. aperiodic: Periodic signals repeat at regular intervals.
• Energy vs. power signals: Energy signals have finite total energy; power signals have finite average power.

System Properties

Key properties characterize signal processing systems:

• Linearity: Output scales proportionally with input; superposition applies.
• Time-invariance: System behavior doesn't change with time.
• Causality: Output depends only on present and past inputs.
• Stability: Bounded inputs produce bounded outputs (BIBO stability).
• Memory: Output depends on input history, not just current value.

Linear time-invariant (LTI) systems are fundamental because they are fully characterized by impulse response or transfer function, enabling powerful analysis techniques.

Convolution

Convolution describes how LTI systems transform inputs. For continuous-time systems:

y(t) = integral of x(tau) h(t - tau) d(tau)

Where h(t) is the system impulse response. Convolution in time domain corresponds to multiplication in frequency domain, a key insight enabling efficient filter design and analysis.

Fourier Transform

The Fourier transform decomposes signals into frequency components:

X(f) = integral of x(t) exp(-j 2 pi f t) dt

Key properties include:

• Linearity: Transform of sum equals sum of transforms.
• Time shifting: Delay introduces linear phase.
• Frequency shifting: Modulation by carrier shifts spectrum.
• Convolution theorem: Convolution becomes multiplication.
• Parseval's theorem: Energy preserved in frequency domain.

Discrete Fourier Transform (DFT)

The DFT computes frequency content of discrete sequences:

X[k] = sum of x[n] exp(-j 2 pi k n / N)

The Fast Fourier Transform (FFT) computes the DFT efficiently in O(N log N) operations, enabling real-time spectral analysis.

Z-Transform

The Z-transform analyzes discrete-time systems:

X(z) = sum of x[n] z^(-n)

Pole-zero locations in the z-plane determine system stability and frequency response. Poles inside the unit circle indicate stability for causal systems.

Laplace Transform

The Laplace transform generalizes Fourier analysis for continuous-time systems, handling transient behavior and enabling s-domain circuit analysis. Transfer functions H(s) characterize system response.

Sampling Theory

The Nyquist-Shannon sampling theorem states that a band-limited signal can be perfectly reconstructed from samples taken at more than twice the highest frequency. Sampling at rate fs:

• Nyquist rate: 2 * fmax (minimum for reconstruction).
• Aliasing: Frequencies above fs/2 fold back, corrupting the signal.
• Anti-aliasing filter: Lowpass filter before sampling removes high frequencies.

Quantization

Quantization maps continuous amplitude values to discrete levels. Effects include:

• Quantization error: Difference between actual and quantized values.
• Signal-to-quantization-noise ratio: Approximately 6.02 dB per bit for uniform quantization.
• Dithering: Adding noise before quantization can improve perceived quality.
• Non-uniform quantization: Companding (mu-law, A-law) improves SNR for varying signal levels.

Oversampling and Noise Shaping

Sampling faster than Nyquist rate spreads quantization noise across wider bandwidth. Noise shaping pushes noise to frequencies outside the band of interest. Combined in delta-sigma converters to achieve high resolution.

Reconstruction

Digital-to-analog conversion requires reconstruction filtering to remove spectral images at multiples of the sampling frequency. Ideal reconstruction uses a sinc function; practical filters approximate this response.

FIR Filters

Finite Impulse Response filters have non-recursive structure:

y[n] = sum of b[k] x[n-k]

FIR advantages include:

• Inherent stability: No feedback means no instability risk.
• Linear phase: Symmetric coefficients provide constant group delay.
• Simple design: Window method, frequency sampling, Parks-McClellan algorithm.

FIR disadvantages include requiring more coefficients than IIR for equivalent selectivity.

IIR Filters

Infinite Impulse Response filters use feedback:

y[n] = sum of b[k] x[n-k] - sum of a[k] y[n-k]

IIR advantages include:

• Efficiency: Fewer coefficients for sharp transitions.
• Analog prototypes: Can transform Butterworth, Chebyshev, elliptic designs.

IIR challenges include potential instability, nonlinear phase, and limit cycle issues in fixed-point implementations.

Filter Design Methods

• Window method: Truncate ideal impulse response with window function.
• Parks-McClellan: Optimal equiripple FIR design.
• Bilinear transform: Convert analog prototype to digital IIR.
• Impulse invariance: Sample analog impulse response.
• Least squares: Minimize error in frequency or time domain.

Multirate Signal Processing

Changing sample rate enables efficient processing:

• Decimation: Reduce sample rate by filtering and downsampling.
• Interpolation: Increase sample rate by upsampling and filtering.
• Polyphase filters: Efficient implementation for rate conversion.
• Filter banks: Divide signal into frequency bands for separate processing.

Adaptive Filtering

Adaptive filters automatically adjust coefficients to optimize performance. Applications include:

• System identification: Model unknown systems.
• Noise cancellation: Remove correlated interference.
• Echo cancellation: Remove acoustic or electrical echoes.
• Channel equalization: Compensate for channel distortion.
• Prediction: Forecast future signal values.

Least Mean Squares (LMS) Algorithm

LMS adjusts coefficients to minimize mean squared error:

w[n+1] = w[n] + mu * e[n] * x[n]

Where mu is the step size controlling adaptation rate versus stability. LMS is simple and widely used but can be slow converging.

Recursive Least Squares (RLS)

RLS converges faster than LMS by using input correlation information but requires more computation. Provides better tracking in non-stationary environments.

Adaptive Beamforming

Antenna arrays use adaptive processing to steer beams toward desired signals while nulling interferers. Algorithms optimize weights based on signal statistics.

Random Processes

Random signals require statistical characterization:

• Mean: Expected value of the process.
• Autocorrelation: Measures similarity at different time delays.
• Power spectral density: Fourier transform of autocorrelation.
• Stationarity: Statistical properties constant over time.
• Ergodicity: Time averages equal ensemble averages.

Detection and Estimation

Fundamental problems in communication receivers:

• Detection: Decide which of several possible signals was transmitted.
• Estimation: Determine parameters of received signals.
• Maximum likelihood: Choose hypothesis maximizing probability of observations.
• MMSE estimation: Minimize mean squared error.

Spectral Estimation

Techniques for estimating power spectrum from finite data:

• Periodogram: FFT-based estimate, high variance.
• Welch method: Averaged periodograms, reduced variance.
• Parametric methods: Model signal as AR, MA, or ARMA process.
• MUSIC, ESPRIT: High-resolution methods for line spectra.

Modulation and Demodulation

Signal processing implements modulation schemes:

• Carrier generation: NCO (numerically controlled oscillator).
• Mixing: Frequency translation to/from baseband.
• Pulse shaping: Spectral containment with raised cosine filters.
• Symbol timing recovery: Sample at optimal instants.
• Carrier recovery: Phase and frequency synchronization.

Equalization

Compensating for channel distortion:

• Zero-forcing: Invert channel response (noise enhancement risk).
• MMSE: Balance noise enhancement and ISI.
• Decision feedback: Use detected symbols to cancel ISI.
• OFDM: Transform channel into parallel flat-fading channels.

Error Correction

Channel coding adds redundancy for error detection and correction:

• Block codes: Reed-Solomon, BCH, LDPC.
• Convolutional codes: Viterbi decoding.
• Turbo codes: Near-Shannon-limit performance.
• Soft decision decoding: Use reliability information.

Spread Spectrum Processing

Spreading and despreading signals:

• PN sequence generation: LFSR-based pseudo-random codes.
• Correlation: Despreading matched to spreading code.
• RAKE receiver: Combine multipath components.
• Acquisition and tracking: Synchronize to spreading code.

DSP Processors

Specialized processors optimize signal processing operations:

• MAC units: Single-cycle multiply-accumulate.
• Circular buffers: Efficient filter implementation.
• Harvard architecture: Separate program and data memory.
• SIMD: Single instruction, multiple data parallelism.

FPGA Implementation

FPGAs enable parallel, pipelined signal processing:

• DSP blocks: Dedicated multiplier-accumulator resources.
• Block RAM: On-chip memory for coefficients and delays.
• Parallel processing: Multiple filters simultaneously.
• Reconfigurability: Update algorithms in field.

Fixed-Point Considerations

Fixed-point arithmetic requires careful attention to:

• Word length: Bits for coefficient and data representation.
• Scaling: Prevent overflow through proper scaling.
• Coefficient quantization: Effects on filter response.
• Round-off noise: Accumulation of quantization errors.
• Limit cycles: Parasitic oscillations in IIR filters.

Audio Processing

Equalization, compression, echo cancellation, noise reduction, spatial audio.

Image and Video Processing

Filtering, enhancement, compression, computer vision algorithms.

Radar and Sonar

Pulse compression, Doppler processing, beamforming, target detection.

Biomedical

ECG/EEG analysis, medical imaging, physiological signal monitoring.

Control Systems

Sensor fusion, state estimation, feedback control implementation.