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信号处理
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Discrete convolution
Discrete convolution
(x∗h)[n]=Σx[k]h[n−k] · 翻转-平移-乘-加 · 逐移位累加
y[n] = Σₖ x[k]·h[n−k] — flip h, slide it across x, multiply and sum
Impulse response of an LTI system
Impulse response of an LTI system
δ[n]→h[n] · LTI输入=移位冲激叠加=移位h叠加
δ[n] → h[n]; any input = Σ scaled·shifted impulses → output = Σ scaled·shifted h
Autocorrelation of a signal
Autocorrelation of a signal
Rxx[k]=Σx[n]x[n+k] · lag0峰 · 周期峰揭示周期
Rₓₓ[k] = Σₙ x[n]·x[n+k] — self-similarity at lag k; peaks reveal the period
Aliasing — when sampling lies
Aliasing — when sampling lies
fs<2f欠采样 · 别名|f−fs·round(f/fs)| · Nyquist fs/2
Sample too slowly and a high tone disguises itself as a low one · f_alias = |f − fs·round(f/fs)|
ADC quantization
ADC quantization
b位=2^b级 · Δ=range/2^b · 误差±Δ/2 · SQNR≈6.02b
b bits → 2^b levels · step Δ = range/2^b · SQNR ≈ 6.02·b + 1.76 dB
Windowing & spectral leakage
Windowing & spectral leakage
矩形vs Hann/Hamming · 截断致泄漏 · 主瓣vs旁瓣
Chopping a signal to finite length smears one tone across many bins · Hann · Hamming · Rectangular
The Fourier transform — magnitude spectrum
The Fourier transform — magnitude spectrum
X[k]=Σx[n]e^(−j2πkn/N) · 多音在分量频率出峰
X[k] = Σₙ x[n]·e^(−j2πkn/N) · |X[k]| reveals which frequencies are present
RC low-pass & high-pass filters
RC low-pass & high-pass filters
LP 1/√(1+(f/fc)²) · HP (f/fc)/√… · −3dB截止
LP |H(f)| = 1/√(1+(f/fc)²) · HP |H(f)| = (f/fc)/√(1+(f/fc)²) · fc = −3 dB point
FIR filter — taps & frequency response
FIR filter — taps & frequency response
y[n]=Σb_k x[n−k] · 抽头系数 · |Σb_k e^{−jωk}|
y[n] = Σₖ bₖ·x[n−k] · |H(e^{jω})| = |Σₖ bₖ·e^(−jωk)|
z-plane poles & stability
z-plane poles & stability
z平面·单位圆 · 极点|z|<1稳定 · 冲激衰减/增长
H(z) is stable ⇔ every pole is inside the unit circle |z| < 1
Amplitude modulation (AM)
Amplitude modulation (AM)
s=(1+m·cos2πf_m t)cos2πf_c t · 载波+边带f_c±f_m
s(t) = (1 + m·cos 2πf_m t) · cos 2πf_c t
Spectrogram (STFT)
Spectrogram (STFT)
滑窗逐段DFT→时频热图 · chirp频率随时间上升
slide a window, take the DFT in each — time × frequency × magnitude