Signal System and Digital Signal Processing

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Unit step u(t): 1 for t≥0, 0 for t<0.

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Unit impulse δ(t): zero everywhere except t=0, with unit area; obeys the sifting property ∫x(t)δ(t−t₀)dt = x(t₀).

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Sinc function: sinc(t) = sin(πt)/(πt); forms the Fourier transform pair of an ideal rectangular (low pass) spectrum.

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Signum function sgn(t): +1 for t>0, −1 for t<0, 0 at t=0.

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Impulse response h(t)/h[n]: the output of a system when the input is a unit impulse — fully characterizes an LTI system.

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Convolution: y(t) = x(t)*h(t) = ∫x(τ)h(t−τ)dτ (continuous) and y[n] = x[n]*h[n] = Σx[k]h[n−k] (discrete) — gives the output of an LTI system for any input.

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Hilbert transform: produces a 90° phase-shifted version of a signal without changing its magnitude spectrum; used to form the analytic signal and in SSB modulation.

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Continuous-Time Fourier Series (CTFS): represents a periodic continuous-time signal as a sum of harmonically related complex exponentials, with coefficients found by integrating over one period.

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Discrete-Time Fourier Series (DTFS): represents a periodic discrete-time signal using a finite sum of N harmonics, unlike CTFS which can have infinite terms; inherently periodic in frequency.

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Continuous-Time Fourier Transform (CTFT): extends Fourier analysis to aperiodic (and periodic, via impulses) signals, X(ω) = ∫x(t)e^(−jωt)dt, giving a continuous frequency spectrum.

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Discrete-Time Fourier Transform (DTFT): frequency-domain representation of a discrete-time sequence, X(e^(jω)) = Σx[n]e^(−jωn), always periodic in ω with period 2π.

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Energy Spectral Density (ESD): distribution of a finite-energy signal's energy over frequency, |X(ω)|².

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Power Spectral Density (PSD): distribution of a power signal's average power over frequency — unlike ESD, which applies to finite-energy signals.

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Ideal sampling models sampling as multiplying a continuous-time signal by a train of impulses spaced T_s apart.

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Sampling theorem: the original signal can be perfectly reconstructed if the sampling rate f_s ≥ 2f_max (the Nyquist rate).

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Linearity: an LTI system obeys superposition — response to a₁x₁+a₂x₂ equals a₁y₁+a₂y₂.

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Time-invariance: a time shift in the input produces an identical time shift in the output, with no other change.

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Causality: output depends only on present and past input values, never future values.

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Stability (BIBO): every bounded input produces a bounded output.

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Memory: output depends on past/future inputs; memoryless systems have output depending only on the present input.

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Frequency response H(ω): the Fourier transform of the impulse response; describes how an LTI system scales and phase-shifts each sinusoidal frequency component of the input.

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Ideal low pass filter: H(ω) = 1 for |ω|≤ω_c and 0 elsewhere (a perfect rectangular passband).

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The ideal LPF's impulse response is a sinc function, and its step response exhibits overshoot/ringing (Gibbs phenomenon) since the ideal filter is non-causal and physically unrealizable.

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BIBO stability condition: an LTI system is BIBO stable if and only if its impulse response is absolutely summable/integrable (Σ|h[n]|<∞ or ∫|h(t)|dt<∞).

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Equivalently, for causal systems described by a transfer function: all poles must lie in the left half s-plane (continuous) or strictly inside the unit circle (discrete, z-plane).

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Implementation of an LTI system: realized via difference equations (discrete) or differential equations (continuous), translated into block diagrams/hardware using delay elements, multipliers, and adders (e.g. direct form realizations for digital filters).

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Z-transform of a sequence x[n] is X(z) = Σx[n]z⁻ⁿ, generalizing the DTFT (which is X(z) evaluated on the unit circle, z=e^(jω)).

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Region of Convergence (ROC): the set of z-values for which the z-transform sum converges; the ROC together with X(z) uniquely determines x[n].

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Causal sequence: has an ROC that is the exterior of a circle (|z|>r); anti-causal sequence: ROC is the interior of a circle.

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Inverse z-transform recovers x[n] from X(z), commonly via partial fraction expansion, power series expansion, or the contour integral (residue) method.

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Convolution property: convolution in the time domain corresponds to multiplication of z-transforms: Zx[n]*h[n] = X(z)H(z).

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Parseval's theorem (z-transform form) relates the energy of a sequence to a contour integral of its z-transform.

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System (transfer) function H(z) = Y(z)/X(z) characterizes an LTI system; evaluating H(z) on the unit circle gives the steady-state sinusoidal (frequency) response, while transient response relates to the system's poles.

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Pole-zero relationship to stability: for a causal system, BIBO stability requires all poles of H(z) to lie strictly inside the unit circle; a pole on or outside the unit circle causes instability (unbounded or sustained oscillatory response).

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DFT: samples the DTFT at N equally spaced frequency points; it is the DTFT of a finite-length sequence evaluated at discrete frequencies, connecting continuous-frequency analysis to a computable, finite set of values.

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Frequency domain sampling: sampling the DTFT in frequency corresponds to periodic repetition of the sequence in time; the original finite-length sequence can be recovered from enough frequency samples.

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Properties of the DFT: linearity, circular time/frequency shift, circular convolution (time-domain multiplication of DFTs), Parseval's theorem (DFT form), symmetry for real sequences.

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Circular convolution: multiplying two sequences' DFTs and taking the inverse DFT yields the circular (not linear) convolution of the original sequences — a key distinction from ordinary (linear) convolution.

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Direct form (FIR/IIR): implements the difference equation directly using delay elements, multipliers by the filter coefficients, and adders.

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Lattice structure: an alternative realization using cascaded stages with reflection coefficients; offers good numerical properties (robust to coefficient quantization) and modularity.

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Lattice-ladder structure (for IIR): extends the lattice structure with an additional ladder part to realize the full IIR transfer function (poles and zeros).

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Direct ↔ lattice conversion: direct-form coefficients can be converted to/from lattice reflection coefficients via recursive algorithms (e.g. the Levinson-Durbin-type recursion).

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Fixed-point representation: a fixed number of bits allocated to the integer and fractional parts; simple hardware, limited dynamic range.

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Floating (binary) point representation: represents numbers as mantissa × 2^exponent; wider dynamic range at the cost of more complex hardware and variable precision.

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Rounding: approximates a value to the nearest representable level, with error bounded symmetrically (±half the quantization step).

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Truncation: simply discards extra bits, giving a biased (one-directional) error rather than the symmetric error of rounding.

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Quantization of filter coefficients: representing filter coefficients with finite precision perturbs the actual pole/zero locations from their ideal design values.

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Poles near the unit circle (high-Q, narrow-band filters) are especially sensitive to coefficient quantization, potentially causing instability or degraded frequency response.

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Classical (analog prototype) IIR design: design a well-known analog filter (Butterworth, Chebyshev, Elliptic) meeting the specification, then transform it to a digital filter (e.g. via bilinear transform).

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Impulse-invariant method: digital filter's impulse response is obtained by sampling the analog prototype filter's impulse response; preserves the time-domain shape but can suffer aliasing in the frequency response.

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IIR low pass discrete filter design: typically starts from a normalized analog low pass prototype, transformed to the digital domain and frequency-scaled to the desired cutoff.

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Fourier approximation (FIR design): directly computes filter coefficients as the (truncated) inverse Fourier transform of the desired frequency response; truncation causes Gibbs phenomenon (ripples) near band edges.

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Window function method: multiplies the truncated ideal impulse response by a window function (Hamming, Hanning, Blackman, Kaiser, etc.) to taper the truncation and reduce ripple/improve stopband attenuation, at the cost of a wider transition band.

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Frequency sampling method: specifies the desired frequency response at a set of discrete frequency points, then computes filter coefficients via the inverse DFT; particularly convenient for designing filters with an arbitrary desired shape.

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Remez exchange algorithm: an iterative optimization algorithm that designs an optimum equi-ripple linear-phase FIR filter, minimizing the maximum error (Chebyshev/minimax criterion) between the actual and desired frequency response.

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Equi-ripple design (via the Remez exchange algorithm, also known as the Parks-McClellan algorithm) distributes the approximation error equally across ripples in both passband and stopband.

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Equi-ripple design achieves the minimum possible filter order for a given set of specifications — generally superior to window-based design for a given order.

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Common window functions used in FIR filter design include the Hamming, Hanning, Blackman, and Kaiser windows.

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Window method trades ripple reduction for a wider transition band, while Remez exchange optimally distributes minimax error for a given filter order.

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A digital filter processes a sampled input sequence according to a difference equation (FIR or IIR).

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Digital filters are realized in hardware (DSP chips, FPGAs, ASICs) or software (running on a general-purpose processor), depending on speed, power, and flexibility requirements.

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Implementation choices (direct form, cascade of second-order sections, parallel form, lattice) trade off computational efficiency, numerical robustness (sensitivity to coefficient quantization), and ease of design.

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FFT (Fast Fourier Transform): an efficient algorithm for computing the DFT, reducing computational complexity from O(N²) for a direct DFT computation to `O(Nlog₂N)`.

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The FFT's O(Nlog₂N) efficiency makes frequency-domain processing of large data sets practical.

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Fast convolution: using the FFT to compute linear/circular convolution efficiently via the convolution theorem (convolution in time = multiplication in frequency), much faster than direct time-domain convolution for long sequences.

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Spectral analysis: the FFT is used to estimate a signal's frequency content, essential in applications such as vibration analysis, audio processing, and communications.

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Filtering in the frequency domain: the FFT enables filtering by multiplying the signal's spectrum with a desired frequency response, then taking the inverse FFT — efficient for high-order filters or block processing.

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Other FFT applications include image processing (2-D FFT), radar/sonar signal processing, and OFDM (Orthogonal Frequency Division Multiplexing) systems in modern digital communications.

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Cascade (second-order sections) implementation is chosen mainly to improve numerical robustness / reduce sensitivity to coefficient quantization compared to a single high-order direct form.

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Hardware implementations (DSP/FPGA/ASIC) trade off against software implementations on general-purpose processors based on speed, power, and flexibility needs.