环形折积

定义

${\displaystyle x_{T}(t)\ {\stackrel {\mathrm {def} }{=}}\ \sum _{k=-\infty }^{\infty }x(t-kT)=\sum _{k=-\infty }^{\infty }x(t+kT).}$

x(t)h(t)计算环形折积的运算(${\displaystyle x(t)\otimes h(t)}$)可以下列两种等价表示式定义:

{\displaystyle {\begin{aligned}y(t)&=\int _{t_{o}}^{t_{o}+T}h_{T}(\tau )\cdot x_{T}(t-\tau )\,d\tau \\&=\int _{-\infty }^{\infty }h(\tau )\cdot x_{T}(t-\tau )\,d\tau \quad {\stackrel {\mathrm {def} }{=}}\quad x_{T}(t)*h(t),\end{aligned}}}

{\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }h(\tau )\cdot x_{T}(t-\tau )\,d\tau &=\sum _{k=-\infty }^{\infty }\left[\int _{t_{o}+kT}^{t_{o}+(k+1)T}h(\tau )\cdot x_{T}(t-\tau )\ d\tau \right]\\&=\sum _{k=-\infty }^{\infty }\left[\int _{t_{o}}^{t_{o}+T}h(\tau +kT)\cdot x_{T}(t-\tau -kT)\ d\tau \right]\\&=\sum _{k=-\infty }^{\infty }\left[\int _{t_{o}}^{t_{o}+T}h(\tau +kT)\cdot x_{T}(t-\tau )\ d\tau \right]\\&=\int _{t_{o}}^{t_{o}+T}\left[\sum _{k=-\infty }^{\infty }h(\tau +kT)\cdot x_{T}(t-\tau )\right]\ d\tau \\&=\int _{t_{o}}^{t_{o}+T}\left[\sum _{k=-\infty }^{\infty }h(\tau +kT)\right]\cdot x_{T}(t-\tau )\ d\tau \\&\ {\stackrel {\mathrm {def} }{=}}\ \int _{t_{o}}^{t_{o}+T}h_{T}(\tau )\cdot x_{T}(t-\tau )\ d\tau \quad \quad {\mbox{QED}}\end{aligned}}}

{\displaystyle {\begin{aligned}x_{N}[n]*h[n]\ &{\stackrel {\mathrm {def} }{=}}\ \sum _{m=-\infty }^{\infty }h[m]\cdot x_{N}[n-m]\\&=\sum _{m=-\infty }^{\infty }h[m]\cdot \sum _{k=-\infty }^{\infty }x[n-m-kN].\,\end{aligned}}}

线性折积与环形折积之比较

线性折积

${\displaystyle (f\star g)(t)=\int f(\tau )g(t-\tau )\,d\tau }$

${\displaystyle (f\star g)[m]=\sum _{n}{f[n]g[m-n]}}$

如何利用快速傅立叶变换计算线性折积

${\displaystyle y[n]=x[n]\star h[n]=\sum _{k}{x[n-k]h[k]}}$
${\displaystyle y[n]=IFFT(FFT{x[n]}FFT{h[n]})}$

${\displaystyle y_{1}[n]=IFFT_{L}(FFT_{L}{x_{1}[n]}FFT_{L}{h_{1}[n]})}$
${\displaystyle y_{1}[n]=\sum _{k=0}^{N-1}{x[((n-k))_{L}]h[k]}}$

${\displaystyle y[n]=x[n]\star h[n]=\sum _{k}{x[n-k]h[k]}}$

x[n] 与 h[n] 皆为有限长信号

${\displaystyle y[n]=\sum _{k=k_{1}}^{k_{2}}{x[n-k]h[k]}}$

${\displaystyle x_{1}[n]=x[n+n_{1}],n=0,1,2,...,N-1}$
${\displaystyle x_{1}[n]=0,n=N,N+1,N+2,...,M-1M=N+K-1}$
${\displaystyle h_{1}[n]=h[n+k_{1}],n=0,1,2,...,K-1}$
${\displaystyle h_{1}[n]=0,n=K,K+1,K+2,...,M-1}$
${\displaystyle y_{1}[n]=IFFT_{M}(FFT_{M}{x_{1}[n]}FFT_{M}{h_{1}[n]})}$
${\displaystyle y[n]=y_{1}[n-n_{1}-k_{1}],n-n_{1}-k_{1}=0,1,2,...,N-1}$

x[n] 为无限长信号，而h[n] 为有限长信号

${\displaystyle x_{1}[n]=x[n+n_{1}],n=0,1,2,...,N-1}$
${\displaystyle x_{1}[n]=0,n=N,N+1,N+2,...,L-1L=N+M-1}$
${\displaystyle h_{1}[n]=h[n+m_{1}-n_{2}],n=0,1,2,...,L-1}$
${\displaystyle y_{1}[n]=IFFT_{L}(FFT_{L}{x_{1}[n]}FFT_{L}{h_{1}[n]})}$
${\displaystyle y[n]=y_{1}[n-m_{1}+N-1],n-m_{1}+N-1=N-1,N,...,L}$ (只选y1[n]后面M个点)

参考

• Rabiner, Lawrence R.; Gold, Bernard. Theory and application of digital signal processing. Englewood Cliffs, N.J.: Prentice-Hall. 1975: pp 63–67. ISBN 0-13-914101-4..
• Oppenheim, Alan V.; Schafer, Ronald W.; Buck, John A. Discrete-time signal processing. Upper Saddle River, N.J.: Prentice Hall. 1999. ISBN 0-13-754920-2..
• Jian-Jiun Ding, Advanced digital signal processing class note,the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.