Signal Processing·Class Hub

Τυπολόγιο

Όσα δεν χρειάζεται να μάθεις απέξω — δίνονται στην εξέταση. Κάθε τύπος συνδέεται με την ενότητα όπου παράγεται, ώστε να βλέπεις πώς προκύπτει αν ξεχάσεις. Άσε σχόλιο σε όποιον τύπο σε δυσκόλεψε.

Foundations

10 τύποι

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

11 τύποι

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

12 τύποι

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

6 τύποι

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

6 τύποι

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}

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Τυπολόγιο · Signal Processing Class Hub