Signals, Systems and Frequency Domain Analysis

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This section covers signal classification, elementary/standard signals, signal transformations, and the properties of LTI systems including convolution.

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Signal Classification Classification Description Continuous-time (CT) vs Discrete-time (DT) CT: defined for every instant t, x(t).

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DT: defined only at integer sample instants n, x[n] Periodic vs Aperiodic Periodic: x(t)=x(t+T) for some T>0.

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Aperiodic: no such T exists Energy vs Power signal Energy signal: finite total energy, zero average power (e.g. a pulse).

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Power signal: finite average power, infinite energy (e.g. periodic signals) Even vs Odd Even: x(−t)=x(t) (symmetric).

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Odd: x(−t)=−x(t) (antisymmetric) Causal / Anticausal / Noncausal Causal: x(t)=0 for t<0.

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Anticausal: x(t)=0 for t>0.

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Noncausal: nonzero for both t<0 and t>0 Any signal decomposes into even + odd parts: x(t) = xₑ(t) + xₒ(t), with xₑ(t) = [x(t)+x(−t)]/2 and xₒ(t) = [x(t)−x(−t)]/2.

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Orthogonal signals: two signals are orthogonal over an interval if the integral of their product (one conjugated, if complex) over that interval is zero.

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Elementary/Standard Signals Signal Definition / Note Unit impulse δ(t) Zero everywhere except t=0, unit area; sifting property: ∫x(t)δ(t−t₀)dt = x(t₀) Unit step u(t) 1 for t≥0, 0 for t<0 Unit ramp r(t) r(t) = t·u(t): equals t for t≥0, 0 for t<0 Real exponential x(t) = Ae^(at); grows if a>0, decays if a<0 Complex exponential x(t) = Ae^(jωt) = A(cosωt + jsinωt) Signum, sgn(t) +1 for t>0, −1 for t<0, 0 at t=0 Transformation of the Independent Variable Time shifting: x(t−t₀) delays the signal if t₀>0, advances it if t₀<0.

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Time scaling: x(at) compresses the signal in time if |a|>1, expands it if |a|<1.

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Time reversal: x(−t) flips the signal about the t=0 axis.

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Combined operations apply scaling/reversal and shifting together, e.g., x(at−b).

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Systems & LTI Properties Property Meaning Linearity Obeys superposition: response to a₁x₁+a₂x₂ equals a₁y₁+a₂y₂ Time- invariance A time shift in the input causes an identical time shift in the output Causality Output at any time depends only on present and past inputs Stability (BIBO) A bounded input always produces a bounded output Memory Output depends on past/future input values (a memoryless system depends only on the present input) An LTI (linear, time-invariant) system is fully characterized by its impulse response h(t) or h[n].

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Convolution integral (CT): y(t) = x(t)*h(t) = ∫x(τ)h(t−τ)dτ — gives the output of an LTI system.

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Convolution sum (DT): y[n] = x[n]*h[n] = Σx[k]h[n−k].

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This section covers the Laplace transform definition and properties, transforms of common functions, the inverse transform and Heaviside's expansion theorem, and its use in solving differential equations and finding transfer functions.

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Definition & Properties Laplace transform:

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X(s) = ∫₀^∞ x(t)e^(−st) dt (unilateral, for t≥0).

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Key properties: linearity, time shifting, frequency shifting, differentiation in time (→ sX(s)−x(0)), integration in time (→ X(s)/s), convolution in time ↔ multiplication in the s-domain, initial-value theorem, final-value theorem.

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Laplace Transforms of Common Functions x(t) X(s) u(t) — unit step 1/s δ(t) — unit impulse 1 e^(−at)u(t) 1/(s+a) t·u(t) — ramp 1/s² sin(ωt)u(t) ω/(s²+ω²) cos(ωt)u(t) s/(s²+ω²) Inverse Laplace Transform & Partial Fractions The inverse Laplace transform recovers x(t) from X(s), usually via partial fraction expansion into simple terms whose inverse transforms are known from tables.

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Heaviside's expansion theorem (the “cover-up” rule) is a systematic method for finding the coefficients (residues) of a partial fraction expansion, especially for simple (non-repeated) poles.

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Solving Differential Equations, Transfer Function & Frequency Response The Laplace transform converts a linear differential equation (in time) into an algebraic equation (in s) — far easier to solve; the solution is then inverse-transformed back to the time domain.

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H(s) = Y(s)/X(s), assuming zero initial conditions — fully characterizes an LTI system.

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H(jω), obtained by substituting s=jω into H(s) — describes the system's steady-state response to sinusoids of different frequencies.

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This section covers the continuous-time Fourier series, the Fourier transform for aperiodic signals and its properties, Parseval's theorem, and the discrete-time Fourier series (DTFS) and discrete-time Fourier transform (DTFT).

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Continuous-Time Fourier Series (CTFS) Any periodic signal (satisfying the Dirichlet conditions, e.g., absolute integrability over one period) can be represented as a weighted sum of harmonically related complex exponentials — the Fourier series.

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Exponential form: x(t) = Σcₖ·e^(jkω₀t), where cₖ are the Fourier series coefficients and ω₀=2π/T is the fundamental frequency.

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Fourier Transform (Aperiodic Signals) The Fourier transform extends Fourier analysis to APERIODIC (and, using impulses, periodic) signals:

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X(jω) = ∫x(t)e^(−jωt)dt (forward); x(t) = (1/2π)∫X(jω)e^(jωt)dω (inverse).

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Key properties: linearity, time shifting, frequency shifting (modulation), time/frequency scaling, differentiation, and convolution (time-domain convolution ↔ frequency-domain multiplication, and vice versa).

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Parseval's theorem: total signal energy is the same whether computed in the time or frequency domain: ∫|x(t)|²dt = (1/2π)∫|X(jω)|²dω.

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Discrete-Time Fourier Series (DTFS) & Transform (DTFT) Transform Applies To Key Feature DTFS Periodic discrete-time signals A FINITE number of harmonics (unlike CTFS, which can have infinitely many) DTFT Aperiodic discrete-time signals Result X(e^(jω)) is CONTINUOUS and PERIODIC in ω (period 2π) — unlike the CT Fourier transform

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This section covers sampling and the sampling theorem, aliasing and reconstruction, the z-transform and its properties, the inverse z-transform, discrete system stability, and the DFT/FFT.

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Sampling & the Sampling Theorem Sampling converts a continuous-time signal into a discrete-time signal by taking values at regular intervals Tₛ (sampling period), fₛ = 1/Tₛ (sampling frequency).

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Nyquist-Shannon sampling theorem: a bandlimited signal (max frequency fₘₐₓ) can be perfectly reconstructed from its samples if fₛ > 2fₘₐₓ (the Nyquist rate).

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Aliasing: distortion that occurs when a signal is sampled BELOW the Nyquist rate — high frequencies are “folded back” and appear as false, lower frequencies.

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Reconstruction from samples typically uses a Zero-Order Hold (ZOH), which holds each sample value constant until the next sample arrives, followed by an anti-imaging low-pass filter.

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Z-Transform Definition & Properties Z-transform:

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X(z) = Σx[n]·z^(−n) — the discrete-time analogue of the Laplace transform.

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Key properties: linearity, time shifting, time-domain convolution ↔ multiplication in the z-domain, scaling in the z-domain.

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The z-transform relates to the DTFT by evaluating X(z) on the unit circle: z = e^(jω).

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Inverse Z-Transform & System Response Inverse z-transform methods: partial fraction expansion, long division (power series), or the contour (inverse) integral.

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Transfer function H(z) = Y(z)/X(z) characterizes a discrete-time LTI system; the poles and zeros of H(z) determine its frequency response and stability.

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Stability, DFT & FFT Stability test: a discrete-time LTI system is BIBO stable if and only if ALL poles of H(z) lie STRICTLY INSIDE the unit circle (|z|<1) — the discrete-time analogue of the left-half-plane condition for continuous systems.

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DFT (Discrete Fourier Transform): computes the frequency-domain representation of a FINITE-length discrete signal, at N equally spaced frequency points.

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FFT (Fast Fourier Transform): a computationally efficient algorithm for the DFT, reducing complexity from O(N²) to O(N·log N).

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This section covers practical applications of frequency-domain techniques: stability and spectral analysis, spectrum sensing, correlation, and frequency-domain system design.

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Stability & Spectral Analysis Frequency-domain techniques (Bode, Nyquist, pole-zero plots) are used for stability analysis of both continuous- and discrete-time systems (as covered in Section 5.6).

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Spectral analysis examines a signal's frequency content (its spectrum) — e.g., via the Fourier transform or DFT/FFT — to identify dominant frequency components, harmonics, or noise.

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Spectrum Sensing & Correlation Spectrum sensing (notably in cognitive radio/communications) detects which parts of the frequency spectrum are currently occupied or free, enabling opportunistic/efficient spectrum use.

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Correlation measures the similarity between two signals (cross-correlation) or between a signal and a shifted version of itself (autocorrelation); widely used for detecting signals in noise and for system identification.

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System Design Using the Frequency Domain Frequency-domain design specifies desired system behaviour (bandwidth, gain, phase margin, filter cutoff) directly in terms of frequency response, then derives a transfer function/filter meeting those specifications — often more intuitive than pure time-domain design.

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This section covers filter fundamentals and types, ideal vs practical filters, analog filter approximations (Butterworth, Chebyshev), and digital FIR/IIR filters.

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Filter Fundamentals Filter Type Passes Low-pass filter (LPF) Frequencies below a cutoff High-pass filter (HPF) Frequencies above a cutoff Band-pass filter (BPF) Frequencies within a specified band Band-stop (notch) filter Rejects frequencies within a band, passes the rest An ideal filter has a perfectly sharp (brick-wall) transition between passband and stopband — NOT physically realizable, only approximated.

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Passive filters: built from R, L, C only, no gain, no external power.

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Active filters: include an amplifying element (op-amp), can provide gain, avoid inductors (see Section 2.5).

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Analog Filter Approximations Filter Characteristic Butterworth Maximally flat magnitude response in the passband (no ripple); moderate roll-off Chebyshev (Type I) Sharper roll-off than Butterworth, but has ripple in the passband Chebyshev (Type II) Flat passband, ripple confined to the stopband Digital Filters — FIR & IIR Type Feature FIR (Finite Impulse Response) Impulse response of finite duration;

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ALWAYS stable; can have exactly linear phase; generally needs a higher order for a given specification IIR (Infinite Impulse Response) Impulse response of infinite duration (uses feedback); meets steep specifications with a lower order, but may be unstable and typically has nonlinear phase A digital filter's transfer function H(z) = Y(z)/X(z) is implemented as a difference equation in discrete time.

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The basic filtering types (low-pass, high-pass, band-pass, band-stop) apply equally to digital filters, realized in the z-domain instead of the s-domain.