Signals and Systems 訊號與系統

Input-Process-Output (IPO) model

Analog Signal Processing System:

Input: Analog Signal
Process: Analog Signal Processor
Output: Analog Signal

Digital Signal Processing System:

Input: Analog Signal
Process: ADC -> Digital Signal Processor -> DAC
Output: Analog Signal

Transfer Function/System Function - used to represent the black box mathematically.
transfer function 轉換函數/轉移函數/傳遞函數
system function 系統函數

represented as H().

And h is the impulse response of the system.

Example:
system function H(z)
impulse response h[n]

For a system H(z) = Y(z) / X(z) => y[n] = h[n] * x[n]
Since the z-transform of an impulse signal is δ(z) = Σδ[n]z^-n = 1,
the impulse response is h[n]:
Y(z) = H(z)X(z) = H(z)δ(z) = H(z) ====z^-1====> h[n]

In terms of Fourier transform, the impulse response is h[n] = F^-1{H(e^jω)}

For an LTI system:

Continuous-Time Signal 連續時間訊號

input - x(t)
output - y(t)
system - H(s) => H(jΩ) (Frequency Domain 頻域)
y(t) = h(t) * x(t)
H(s) = Y(s)/X(s)

Discrete-Time Signal 離散時間訊號

input - x[n]
output - y[n]
system - H(z) => H(e^jω) (Frequency Domain 頻域)
y[n] = h[n] * x[n]
H(z) = Y(z)/X(z)

Discrete-time Signal Processing (2nd edition) by A. Oppenheim:
2.1 Basic Sequences and Sequence Operations

Unit sample sequence: 𝛿[n]
Unit step sequence: u[n]
Exponential sequence: x[n] = Aαⁿ
Sinusoidal sequence: x[n] = Acos(ω₀n+φ)

impulse function 脈衝函數
delta function δ函數

Transfer function H(e^jω)
- mathematical expression for a system
- can be used to derive frequency response including (1) magnitude response (2) phase response
Diagram of (1) and (2) is called the Bode plot 波德圖

impulse response 脈衝響應
frequency response 頻率響應

magnitude response 振幅響應
gain 增益

phase response 相位響應
phase shift 相移/相位偏移

if magnitude and phase output of a system is not changed as desired, there are:
magnitude distortion 振幅失真
phase distortion 相位失真

group delay 群延/群延遲/群組延遲/群體延遲

analytical signal 分析訊號/解析訊號 - complex function without negative frequency.