Materiale completo di ELECTROMAGNETICS AND SIGNAL PROCESSING FOR SPACEBORNE APPLICATIONS per il corso di Aerospace Engineering presso Politecnico di Milano. Il documento tratta: Signal Processing For Spaceborne Application Lecture Notes – Academic Year 2025/26 Aayush Golwala Signal Theory & Processing Contents 1 Lecture 1 – Introduction to Signals 3 1.1 What is a Signal? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2…
Materiale completo di ELECTROMAGNETICS AND SIGNAL PROCESSING FOR SPACEBORNE APPLICATIONS per il corso di Aerospace Engineering presso Politecnico di Milano. Il documento tratta: Signal Processing For Spaceborne Application Lecture Notes – Academic Year 2025/26 Aayush Golwala Signal Theory & Processing Contents 1 Lecture 1 – Introduction to Signals 3 1.1 What is a Signal? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2…
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Signal Processing For Spaceborne Application Lecture Notes – Academic Year 2025/26 Aayush Golwala
Signal Theory & Processing 1.4 Digital Signals Definition 1.4 (Digital Signal). A discrete signal whose samples are represented using a finite number of digits (bits). • Necessary for storage and processing. • Introduces an approximation error (quantisation). Example Quantisation V0 = 3.6026809156 . . . −−−−−−−−→ Q(V0 ) = 3.602. The quantisation error εq = V0 − Q(V0 ) is equivalent to a disturbance (quantisation noise). In this course we assume that the recorded samples are represented with enough digits (bits) to make quantisation noise negligible with respect to other disturbance factors – a reasonable assumption for modern spaceborne systems. 1.5 Deterministic vs. Random Signals We distinguish between deterministic and random signals depending on the possibility to predict signal behaviour in the future based on an observation over a suitable time window. Definition 1.5 (Deterministic Signal). A signal that can be fully characterised by a limited set of parameters: • expressed in closed form: x(t) = sin(2πfc t + φ); ∂2E 1 ∂2E • determined by some equation: e.g. the wave equation − = −ρ(t, x); ∂x2 c2 ∂t2 • described by some algorithm. Definition 1.6 (Random Signal / Stochastic Process). A signal that cannot be fully predicted: • The signal carrying a voice (you cannot predict what someone is about to say). • Noise / unexpected disturbances. • In general, any signal generated by an unknown source. For Remote Sensing applications, the treatment of random signals is fundamental to: 1. Design sensors that are adapted to the characteristics of the signals we plan to acquire. 2. “Clean up” a signal affected by disturbances. 3. Evaluate the accuracy of our measurements (or estimates). 1.6 Examples of Deterministic Signals 1.6.1 Pulses A pulse is a signal with limited…
Signal Theory & Processing Key Result For Linear Time-Invariant (LTI) systems: N X N X xn (t) → yn (t) ⇒ an xn (t − τn ) → an yn (t − τn ). n=1 n=1 3.4 Pulse Response and Convolution Definition 3.3 (Pulse Response). The pulse response h(t) (or hn ) of a system is the output signal produced when the input is a delta (Dirac or Kronecker). LTI systems are completely characterised by their pulse response: knowing h is sufficient to compute the output for any input. 3.4.1 Derivation – Discrete Time h n δn −→ hn (pulse response) h n δn−k −→ hn−k (by time-invariance) hn xk δn−k −→ xk hn−k (by linearity) hn X X xk δn−k −→ xk hn−k (by linearity) k k Since xn = k xk δn−k , we have: P Key Result ∞ X yn = xk hn−k =: xn ∗ hn (convolution sum) k=−∞ 3.4.2 Derivation – Continuous Time By the same reasoning with x(t) = x(τ ) δ(t − τ ) dτ : R Key Result Z ∞ y(t) = x(τ ) h(t − τ ) dτ =: x(t) ∗ h(t) (convolution integral) −∞ 3.5 Properties of Convolution 3.5.1 Duration The total duration of the output signal cannot exceed the sum of durations Ty ≤ Tx + Th . In the discrete-time case: Ny ≤ Nx + Nh − 1. 3.5.2 Area Z Ay = y(t) dt = Ax · Ah . Discrete-time: n hn . P P P n yn = n xn · 10
Signal Theory & Processing g) Integration: dx Y (f ) y(t) = ⇒ X(f ) = for f ̸= 0. dt j2πf h) Duality: sin(πf ) sin(πtB) F f x(t) = rect(t) ⇒ X(f ) = =⇒ −−→ rect . πf πt B i) Delta-like physical signal: δ(t) is an all-pass filter. For any band-limited signal x(t) with total R bandwidth Bx , any signal d(t) such that D(f ) = 1 ∀f ∈ Bx works as a delta: x(t) = x(τ ) d(t − τ ) dτ . j) Frequency shift: n o F x(t) · ej2πf0 t = X(f − f0 ). k) Modulation: F{x(t) · y(t)} = X(f ) ∗ Y (f ). l) Parseval’s Theorem: Z Z x(t) · y ∗ (t) dt = X(f ) · Y ∗ (f ) df. Corollary – energy of a signal: Z Z 2 Ex = |x(t)| dt = |X(f )|2 df. 5.2 Finite Observation Time and Spectral Leakage When observing a signal over a finite time Tobs : n t o sin(πf Tobs ) XTobs (f ) = F x(t) · rect Tobs = X(f ) ∗ . πf The sinc function acts as a spectral spreading kernel of width ≈ 1/Tobs , creating spectral leakage. 5.3 Frequency Resolution F acts as a frequency detector: two sinusoids at fA and fB can be distinguished as long as |fA − fB | > δf = 1/Tobs (frequency resolution). If |fA − fB | < δf , the two peaks merge (constructive or destructive interference depending on phase difference). 6 Lecture 6 – Spectral Density and Correlations 6.1 Energy and Energy Spectral Density By Parseval: Ex = |x(t)|2 dt = |X(f )|2 df . R R The energy carried within a small bandwidth δf centred at f0 : Z f0 +δf /2 δf →0 Ex (f0 , δf ) = |X(f )|2 df −−−→ |X(f0 )|2 · δf. f0 −δf /2 15
Signal Theory & Processing Note Suspicion flag: Be suspicious when a numerical FT looks too “clean” in the interval (−1/2T, 1/2T ); no margins might indicate overlapping replicas! Appendix – Summary of Key Formulae Formula Description z(t) = ρ(t) ejφ(t) Complex signal (polar form) Ex = |x(t)|2 dt Signal energy R 1 |x(t)|2 dt Average power R Px = lim Tobs Discrete delta representation P xn = k xk δn−k Dirac delta representation R x(t) = x(τ ) δ(t − τ ) dτ 1 δ(at) = |a| δ(t) Scaling of Dirac’s delta Discrete convolution P yn = k xk hn−k = xn ∗ hn Continuous convolution R y(t) = x(τ ) h(t − τ ) dτ = x(t) ∗ h(t) X(f ) = x(t) e−j2πf t dt Fourier Transform R x(t) = X(f ) e+j2πf t df Inverse Fourier Transform R F{rect(t/T )} = T · sinc(f T ) FT of rectangular pulse F{x(t) ∗ h(t)} = X(f ) · H(f ) Convolution theorem F{x(t − τ )} = X(f ) e−j2πf τ Delay property Ex = |X(f )|2 df Parseval / Energy spectral density R Rx (t) = x(t) ∗ x∗ (−t) Autocorrelation s(t) = Re{z(t) ej2πf0 t } RF signal – complex envelope fs ≥ B Sampling Theorem condition 1 P Xs (f ) = T k X(f − k/T ) Spectrum of sampled signal Nf ≥ N Condition: no temporal aliasing 20
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