Signal processing Study Guide

📖 Core Concepts Signal – a function of time, \(x(t)\), that may be continuous (analog) or discrete (samples). Analog signal – continuous‑time, not yet digitized; processed with analog circuits. Discrete‑time signal – defined only at specific time instants (quantized in time). Digital signal – numeric samples of a discrete‑time signal; processed by computers/DSPs. Linear Time‑Invariant (LTI) system – described by an impulse response; output = input convolved with impulse response. Transform theory – Fourier, Laplace, and \(z\)-transforms move a signal between time and frequency (or complex‑frequency) domains for analysis/design. Filter – device (analog or digital) that shapes a signal’s spectrum; can be FIR, IIR, or stochastic. Spectral density estimation – determines how signal power is distributed across frequency; reveals hidden components. Statistical signal processing – treats signals as stochastic processes; uses probability to model noise, detect weak components, and estimate parameters. --- 📌 Must Remember Signal types: Analog ↔ Continuous‑time; Discrete‑time ↔ Sampled; Digital ↔ Quantized samples. Sampling theorem (implicit): To avoid aliasing, sampling rate must exceed twice the highest signal frequency. Fourier transform → frequency content; Laplace transform → complex‑frequency (stability analysis); \(z\)-transform → discrete‑time frequency analysis. FIR vs IIR filters: FIR = finite impulse response (always stable, linear phase possible); IIR = infinite impulse response (uses feedback, can be more efficient but may be unstable). Wiener filter – optimal linear filter that minimizes mean‑square error between desired signal and noisy observation. Detection vs Estimation: Detection decides presence of a signal; Estimation finds parameter values (e.g., frequency, amplitude). --- 🔄 Key Processes Sampling & ADC Sample analog signal at \(fs\) Hz → discrete‑time sequence. Quantize each sample → digital numbers (ADC). Designing an LTI Filter (continuous‑time) Write differential equation → derive impulse response \(h(t)\). Convolve input \(x(t)\) with \(h(t)\): \(y(t)=x(t)h(t)\). Designing a Digital FIR Filter Choose desired frequency response. Compute coefficients \(h[n]\) (e.g., window method). Apply convolution sum: \(y[n]=\sum{k=0}^{N-1} h[k]\,x[n-k]\). Spectral Density Estimation Compute Fourier transform of signal or use periodogram. Identify peaks → hidden frequency components. Wiener Filter Computation (optimal linear MMSE) Obtain signal and noise power spectra: \(Sx(f)\), \(Sn(f)\). Form filter transfer function: $$ H{\text{Wiener}}(f)=\frac{Sx(f)}{Sx(f)+Sn(f)} $$ Apply in frequency domain, then inverse‑transform to time domain. --- 🔍 Key Comparisons Analog vs Digital Analog: continuous amplitude, processed with passive/active circuits. Digital: discrete amplitude, processed with DSPs, can implement complex algorithms. Continuous‑time vs Discrete‑time Processing Continuous: uses differential equations, Laplace transform. Discrete: uses difference equations, \(z\)-transform, FIR/IIR filters. FIR vs IIR Filters FIR: finite impulse response, inherently stable, can have exact linear phase. IIR: infinite impulse response, uses feedback, can achieve sharper specs with fewer coefficients but may be unstable. Statistical vs Deterministic Processing Deterministic: assumes exact signal model, uses convolution, transforms. Statistical: models signal/noise as random processes, uses probability, Wiener/ML estimators. --- ⚠️ Common Misunderstandings “Sampling eliminates the need for anti‑aliasing filters.” Wrong: before sampling you must low‑pass filter to limit bandwidth; otherwise high‑frequency energy folds into baseband. “All digital filters are FIR.” Wrong: digital filters can be FIR or IIR; each has trade‑offs. “Fourier transform works on any signal.” It assumes stationary (time‑invariant statistics). Non‑stationary signals need time‑frequency analysis (e.g., STFT, wavelets). “Wiener filter always removes all noise.” It minimizes MSE, not necessarily eliminates noise completely; performance limited by signal‑to‑noise ratio and accurate spectral knowledge. --- 🧠 Mental Models / Intuition Signal ↔ Vector: Treat a finite‑length signal as a point in a high‑dimensional vector space; filtering = projecting onto a subspace defined by the filter’s impulse response. Frequency domain as “color palette”: Each frequency component is a “color” in the signal; filters act like color filters, letting some hues pass while blocking others. Noise as “static” overlay: Detection theory asks “Can we see the picture through the static?” – you set a decision threshold based on acceptable false‑alarm rate. --- 🚩 Exceptions & Edge Cases Non‑linear systems: Linear techniques (Fourier, convolution) fail to capture bifurcations, chaos, or harmonic generation. Finite‑precision arithmetic: In DSPs, rounding errors can destabilize IIR filters; use scaling or fixed‑point analysis. Sparse spectral lines: Standard periodogram may miss weak tones; use higher‑resolution methods (e.g., least‑squares spectral analysis). --- 📍 When to Use Which Choose analog filtering when the signal never leaves the analog domain or when latency must be minimal (e.g., RF front‑end). Use digital FIR for linear‑phase requirements (e.g., audio equalization). Use digital IIR when sharp roll‑off is needed with few coefficients (e.g., communication channel equalization). Apply Wiener filter when you have reliable estimates of signal and noise power spectra and want minimum‑MSE reconstruction. Select statistical methods (e.g., detection theory) when the signal is buried in stochastic noise and you need probabilistic performance guarantees. --- 👀 Patterns to Recognize Peak in spectral density → hidden sinusoidal component. Sharp roll‑off in magnitude response → likely IIR filter. Linear‑phase magnitude curve → FIR filter with symmetric coefficients. Increasing error with higher filter order → possible numerical instability (finite‑precision). --- 🗂️ Exam Traps “All filters are implemented digitally.” – Many problems explicitly ask for analog filter design (use Laplace, continuous‑time impulse response). Confusing sampling rate with Nyquist frequency – Nyquist = \(fs/2\); answer choices often swap them. Choosing Fourier transform for non‑stationary signals – Correct answer is a time‑frequency method (STFT, wavelet). Assuming Wiener filter eliminates noise completely – The trap is to pick “zero error” options; remember it only minimizes MSE. Mixing up FIR and IIR stability statements – FIR is always BIBO stable; IIR may not be. ---