Signal System and Digital Signal Processing

1

This section covers signal fundamentals (impulse response, convolution, elementary functions), the Fourier series and Fourier transform for continuous and discrete time, and energy/power spectral density.

2

Elementary Signals & Convolution Function Description Unit step u(t) 1 for t≥0, 0 for t<0 Unit impulse δ(t) Zero everywhere except t=0, unit area; sifting property ∫x(t)δ(t−t₀)dt=x(t₀) Sinc function sinc(t) = sin(πt)/(πt); the Fourier transform pair of an ideal rectangular (low pass) spectrum Signum function sgn(t) +1 for t>0, −1 for t<0, 0 at t=0 Impulse response h(t)/h[n]: the output of a system when the input is a unit impulse — fully characterizes an LTI system.

3

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

4

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.

5

Fourier Series Representation Key Property Continuous-Time Fourier Series (CTFS) Represents a periodic continuous-time signal as a sum of harmonically related complex exponentials; coefficients found by integrating over one period 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 Fourier Transform & Spectral Density 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.

6

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

7

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

8

Power Spectral Density (PSD): distribution of a power signal's average power over frequency.

1

This section covers ideal sampling, the defining properties of LTI systems, frequency response, the ideal low pass filter's impulse/step response, stability, and practical implementation of LTI systems.

2

Ideal Sampling Ideal sampling models sampling as multiplying a continuous-time signal by a train of impulses spaced Tₛ apart; per the sampling theorem, the original signal can be perfectly reconstructed if the sampling rate fₛ ≥ 2fₘₐₓ (the Nyquist rate).

3

LTI System Properties Property Meaning Linearity Obeys superposition: response to a₁x₁+a₂x₂ equals a₁y₁+a₂y₂ Time-invariance A time shift in the input produces an identical time shift in the output, with no other change Causality Output depends only on present and past input values, never future values Stability (BIBO) Every bounded input produces a bounded output Memory Output depends on past/future inputs (memoryless: output depends only on the present input) Frequency Response & Ideal Low Pass Filter 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.

4

The ideal low pass filter has H(ω) = 1 for |ω|≤ω_c and 0 elsewhere (a perfect rectangular passband); its 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.

5

Stability & Implementation An LTI system is BIBO stable if and only if its impulse response is absolutely summable/integrable (Σ|h[n]| <∞ or ∫|h(t)|dt<∞); 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).

6

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).

1

This section covers the z-transform definition, region of convergence and its link to causality/stability, the inverse z-transform, pole-zero analysis, and the discrete Fourier transform (DFT) and circular convolution.

2

Z-Transform Fundamentals The 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)).

3

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].

4

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

5

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

6

Convolution property: convolution in the time domain corresponds to multiplication of z-transforms:

7

Zx[n]*h[n] = X(z)H(z).

8

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

9

The 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.

10

Pole-Zero Analysis & Stability 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).

11

Discrete Fourier Transform (DFT) Concept Description DFT definition & relation to other transforms The 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 Frequency domain sampling & reconstruction 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 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 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

1

This section covers structural realizations of FIR and IIR filters (direct form, lattice), finite-precision (fixed- point/floating-point) number representation, and the effects of rounding, truncation, and coefficient quantization on filter behavior.

2

FIR & IIR Structures Structure Description Direct Form (FIR/IIR) Implements the difference equation directly using delay elements, multipliers by the filter coefficients, and adders Lattice structure An alternative realization using cascaded stages with reflection coefficients; offers good numerical properties (robust to coefficient quantization) and modularity Lattice-ladder structure (for IIR) Extends the lattice structure with an additional ladder part to realize the full IIR transfer function (poles and zeros) Conversion (direct ↔ lattice) Direct-form coefficients can be converted to/from lattice reflection coefficients via recursive algorithms (e.g. the Levinson-Durbin-type recursion) Finite Precision Number Representation Format Description Fixed-point representation A fixed number of bits allocated to the integer and fractional parts; simple hardware, limited dynamic range Floating (binary) point representation Represents numbers as mantissa × 2^exponent; wider dynamic range at the cost of more complex hardware and variable precision Effects of Finite Precision Rounding: approximates a value to the nearest representable level, with error bounded symmetrically (±half the quantization step).

3

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

4

Quantization of filter coefficients: representing filter coefficients with finite precision perturbs the actual pole/zero locations from their ideal design values — poles near the unit circle (high-Q, narrow-band filters) are especially sensitive, potentially causing instability or degraded frequency response.

1

This section covers classical and impulse-invariant methods for IIR filter design, and several approaches to FIR filter design including window functions, frequency sampling, and the Remez exchange (equi-ripple) algorithm.

2

IIR Filter Design Method Description Classical (analog prototype) 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) 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 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 FIR Filter Design Method Description Fourier approximation Directly computes filter coefficients as the (truncated) inverse Fourier transform of the desired frequency response; truncation causes Gibbs phenomenon (ripples) near band edges 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 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 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 Optimum Equi-Ripple Linear-Phase FIR Filters 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, achieving the minimum possible filter order for a given set of specifications — generally superior to window-based design for a given order.

1

This final section of the chapter covers practical implementation of digital filters and the applications of the Fast Fourier Transform (FFT) algorithm in digital signal processing.

2

Implementation of Digital Filters A digital filter processes a sampled input sequence according to a difference equation (FIR or IIR), realized in hardware (DSP chips, FPGAs, ASICs) or software (running on a general-purpose processor), depending on speed, power, and flexibility requirements.

3

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.

4

Applications of FFT Algorithms The FFT (Fast Fourier Transform) is an efficient algorithm for computing the DFT, reducing computational complexity from O(N²) for a direct DFT computation to O(Nlog₂N), making frequency-domain processing of large data sets practical.

5

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.

6

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

7

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.

8

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