Signal Processing·Class Hub

Τυπολόγιο

Foundations

Rect ↔ sinc

Fourier transform

Π (t / T) \leftrightarrow T sinc (f T)

Triangle ↔ sinc²

Fourier transform

Λ (t / T) \leftrightarrow T sinc^{2} (f T)

Cosine ↔ impulses

Fourier transform

cos (2 π f_{0} t) \leftrightarrow \frac{1}{2} [δ (f - f_{0}) + δ (f + f_{0})]

Sine ↔ impulses (imaginary)

Fourier transform

sin (2 π f_{0} t) \leftrightarrow \frac{1}{2 j} [δ (f - f_{0}) - δ (f + f_{0})]

Scaling

Fourier transform

x (α t) \leftrightarrow \frac{1}{∣ α ∣} X (\frac{f}{α})

Time shift

Fourier transform

x (t - t_{0}) \leftrightarrow X (f) e^{- j 2 π f t_{0}}

Convolution ↔ multiplication

Fourier transform

x (t) * h (t) \leftrightarrow X (f) H (f), x (t) y (t) \leftrightarrow X (f) * Y (f)

Modulation theorem

Fourier transform

x (t) cos (2 π f_{c} t) \leftrightarrow \frac{1}{2} [X (f - f_{c}) + X (f + f_{c})]

Parseval

Fourier transform

\int ∣ x (t) ∣^{2} d t = \int ∣ X (f) ∣^{2} df

Power signal Parseval (cosines)

P_{x} = i \sum \frac{A _{i}^{2}}{2} (για cosines σε διαφορετικ \overset{ε}{ˊ} ς f)

AM

AM σήμα

Conventional AM

x_{A M} (t) = [A_{c} + m (t)] cos (2 π f_{c} t)

Modulation index μ

Conventional AM

μ = \frac{∣ m ∣ _{m a x}}{A _{c}}, μ \leq 1 για να μην υπ \overset{α}{ˊ} ρχει overmodulation

AM φάσμα

Conventional AM

X_{A M} (f) = \frac{A _{c}}{2} [δ (f - f_{c}) + δ (f + f_{c})] + \frac{1}{2} [M (f - f_{c}) + M (f + f_{c})]

AM ισχύς

Conventional AM

P_{A M} = \frac{A _{c}^{2}}{2} + \frac{P _{m}}{2}, P_{m} = \overline{m^{2}}

AM efficiency η

Conventional AM

η = \frac{P _{m}}{A _{c}^{2} + P _{m}} = \frac{μ ^{2} P _{m}}{2 + μ ^{2} P _{m}} \leq \frac{1}{3}

DSB-SC σήμα

x_{D S B - S C} (t) = A_{c} m (t) cos (2 π f_{c} t)

DSB-SC ισχύς

P_{D S B - S C} = \frac{A _{c}^{2} P _{m}}{2}, η = 100%

SSB σήμα

x_{S S B} (t) = A_{c} m (t) cos (2 π f_{c} t) \mp A_{c} \overset{m}{^} (t) sin (2 π f_{c} t)

Hilbert transform

Bandpass & I/Q canonical form

F {\overset{m}{^} (t)} = - j sgn (f) M (f)

AM bandwidth

B_{A M} = B_{D S B - S C} = 2 W, B_{S S B} = W

Envelope detector RC range

AM Modulator/Demodulator

\frac{1}{f _{c}} ≪ R C ≪ \frac{1}{W}

FM / PM

FM σήμα (general)

x_{F M} (t) = A_{c} cos [2 π f_{c} t + 2 π K_{f} \int_{- \infty}^{t} m (τ) d τ]

FM single-tone

x_{F M} (t) = A_{c} cos [2 π f_{c} t + β sin (2 π f_{m} t)]

PM σήμα

x_{P M} (t) = A_{c} cos [2 π f_{c} t + K_{p} m (t)]

Στιγμιαία συχνότητα

f (t) = \frac{1}{2 π} \frac{d θ ( t )}{d t} = f_{c} + K_{f} m (t)

Modulation index β (FM/PM)

β_{f} = \frac{Δ f}{W} = \frac{K _{f} max ∣ m ∣}{W}, β_{p} = K_{p} max ∣ m ∣

Bessel expansion (Jacobi-Anger)

Bessel sidebands

e^{j β s i n θ} = n = - \infty \sum \infty J_{n} (β) e^{j n θ}

FM Bessel form

Bessel sidebands

x_{F M} (t) = A_{c} n = - \infty \sum \infty J_{n} (β) cos [2 π (f_{c} + n f_{m}) t]

Bessel symmetry + power

Bessel sidebands

J_{- n} (β) = (- 1)^{n} J_{n} (β), n = - \infty \sum \infty J_{n}^{2} (β) = 1

Carson's rule

B_{F M} = 2 (Δ f + W) = 2 (β + 1) W

FM ισχύς

Bessel sidebands

P_{F M} = \frac{A _{c}^{2}}{2} (ανεξ \overset{α}{ˊ} ρτητο του β)

FM output SNR

SNR_{o u t, F M} = 3 β^{2} SNR_{r e f} = 3 β^{2} \cdot \frac{A _{c}^{2}}{2 N _{0} W}

FM gain over AM (μ=1)

G_{F M / A M} = 9 β^{2}

Random Processes

Mean function

Random processes

μ_{X} (t) = E [X (t)]

Autocorrelation

Random processes

R_{X} (t_{1}, t_{2}) = E [X (t_{1}) X (t_{2})]

Cross-correlation / cross-covariance

Random processes

R_{X Y} (t_{1}, t_{2}) = E [X (t_{1}) Y (t_{2})], C_{X Y} = R_{X Y} - μ_{X} μ_{Y}

WSS conditions

Stationarity & ergodicity

μ_{X} (t) = μ_{X} = const, R_{X} (t_{1}, t_{2}) = R_{X} (τ)

Wiener-Khinchin (PSD)

S_{X} (f) = F {R_{X} (τ)}, P_{X} = R_{X} (0) = \int S_{X} (f) df

Random-phase cosine

Random processes

X = A cos (2 π f_{0} t + Θ), Θ \sim U [0, 2 π] \Rightarrow μ_{X} = 0, R_{X} (τ) = \frac{A ^{2}}{2} cos (2 π f_{0} τ)

Noise

Thermal noise (Johnson-Nyquist)

P_{N} = k T B, k = 1.38 \times 1 0^{- 23} J/K, T_{0} = 290 K

White noise PSD

S_{N} (f) = \frac{N _{0}}{2}, R_{N} (τ) = \frac{N _{0}}{2} δ (τ), N_{0} = k T

PSD μέσα από LTI

Noise through filters

S_{Y} (f) = ∣ H (f) ∣^{2} S_{X} (f), P_{Y} = \int ∣ H (f) ∣^{2} S_{X} (f) df

Bandpass white noise R(τ)

Noise through filters

R_{Y} (τ) = N_{0} W sinc (W τ) cos (2 π f_{c} τ)

SNR (linear and dB)

SNR = \frac{P _{s}}{P _{n}}, SNR_{d B} = 10 lo g_{10} (SNR)

Noise figure / equivalent T_e

F = \frac{SNR _{in}}{SNR _{o u t}}, T_{e} = (F - 1) T_{0}

Πίσω στο Practice hub

Quiz mode

Σωστό/Λάθος και πολλαπλής επιλογής, με εξήγηση και link στις σχετικές ενότητες θεωρίας μετά από κάθε απάντηση. Άνοιξε το τυπολόγιο όποτε θέλεις από το κουμπί κάτω αριστερά.

1. Επιλέξε topics

Άσε όλα ενεργά για ανάμεικτο quiz, ή φίλτραρε για να εξασκηθείς σε ένα συγκεκριμένο κεφάλαιο.

25 ερωτήσεις στη λίστα.

2. Επιλέξε mode

Quiz — Σωστό/Λάθος + πολλαπλής επιλογής · Signal Processing Class Hub