双采样率下的非均匀采样
假定一个函数g(t)g(t)g(t)的傅里叶变换为
G(f)=∫−∞∞g(t)e−j2πftdt
G(f)=\int_{-\infty}^{\infty}g(t)e^{-j2\pi ft}dt
G(f)=∫−∞∞g(t)e−j2πftdt
定义一个门函数h1(t)h_1(t)h1(t),再定义一个与之互补的门函数h2(t)h_2(t)h2(t),有
h2(t)=1−h1(t)
h_2(t)=1-h_1(t)
h2(t)=1−h1(t)
可见h1(t)h2(t)=0h_1(t)h_2(t)=0h1(t)h2(t)=0,其傅里叶变换分别为:
h1(t)↔H1(f)h2(t)↔H2(f)=δ(f)−H1(f)
h_1(t)\leftrightarrow H_1(f)\\
h_2(t)\leftrightarrow H_2(f)=\delta(f)-H_1(f)
h1(t)↔H1(f)h2(t)↔H2(f)=δ(f)−H1(f)
将它们分别与g(t)g(t)g(t)相乘,得到两个子信号:
w1(t)=g(t)h1(t)w2(t)=g(t)h2(t)
w_1(t)=g(t)h_1(t)\\
w_2(t)=g(t)h_2(t)
w1(t)=g(t)h1(t)w2(t)=g(t)h2(t)
现在,以采样率fs1f_{s1}fs1对w1(t)w_1(t)w1(t)进行采样,以采样率fs2f_{s2}fs2对w2(t)w_2(t)w2(t)进行采样,得到:
ws1(t)=w1(t)∑n1δ(t−nts1)ws2(t)=w2(t)∑n2δ(t−nts2)
w_{s1}(t)=w_1(t)\sum_{n1}\delta(t-nt_{s1})\\
w_{s2}(t)=w_2(t)\sum_{n2}\delta(t-nt_{s2})
ws1(t)=w1(t)n1∑δ(t−nts1)ws2(t)=w2(t)n2∑δ(t−nts2)
其中,n1,n2n_1,n_2n1,n2为对应的采样点数,ts1=1fs1t_{s1}=\frac{1}{f_{s1}}ts1=fs11,ts2=1fs2t_{s2}=\frac{1}{f_{s2}}ts2=fs21.
我们知道:
∑nδ(t−nts)↔fs∑uδ(f−ufs)
\sum_{n}\delta(t-nt_{s})\leftrightarrow f_s\sum_{u}\delta(f-uf_s)
n∑δ(t−nts)↔fsu∑δ(f−ufs)
两部分采样子信号的傅里叶变换可以分别写为:
Ws1(f)=W1(f)∗fs1∑u1δ(f−ufs1)=G(f)∗H1(f)∗fs1∑u1δ(f−ufs1)Ws2(f)=W2(f)∗fs2∑u2δ(f−ufs2)=G(f)∗H2(f)∗fs2∑u2δ(f−ufs2)
W_{s1}(f)=W_1(f)*f_{s1}\sum_{u_1}\delta(f-uf_{s1})=G(f)*H_1(f)*f_{s1}\sum_{u_1}\delta(f-uf_{s1})\\
W_{s2}(f)=W_2(f)*f_{s2}\sum_{u_2}\delta(f-uf_{s2})=G(f)*H_2(f)*f_{s2}\sum_{u_2}\delta(f-uf_{s2})
Ws1(f)=W1(f)∗fs1u1∑δ(f−ufs1)=G(f)∗H1(f)∗fs1u1∑δ(f−ufs1)Ws2(f)=W2(f)∗fs2u2∑δ(f−ufs2)=G(f)∗H2(f)∗fs2u2∑δ(f−ufs2)
现在想恢复出G(f)G(f)G(f),进而恢复出g(t)g(t)g(t),我们只保留其基带成分:
Ws1′(f)=G(f)∗H1(f)∗fs1δ(f)=G(f)∗fs1H1(f)Ws2′(f)=G(f)∗H2(f)∗fs2δ(f)=G(f)∗fs2H2(f)
W'_{s1}(f)=G(f)*H_1(f)*f_{s1}\delta(f)=G(f)*f_{s1}H_1(f)\\
W'_{s2}(f)=G(f)*H_2(f)*f_{s2}\delta(f)=G(f)*f_{s2}H_2(f)
Ws1′(f)=G(f)∗H1(f)∗fs1δ(f)=G(f)∗fs1H1(f)Ws2′(f)=G(f)∗H2(f)∗fs2δ(f)=G(f)∗fs2H2(f)
现构造一个Ws1′(f)W'_{s1}(f)Ws1′(f)和Ws2′(f)W'_{s2}(f)Ws2′(f)的线性组合,令
G(f)=k1Ws1′(f)+k2Ws2′(f)
G(f)=k_1W'_{s1}(f)+k_2W'_{s2}(f)
G(f)=k1Ws1′(f)+k2Ws2′(f)
可得
G(f)=G(f)∗(k1fs1H1(f)+k2fs2H2(f))=G(f)∗(k2fs2δ(f)+(k1fs1−k2fs2)H1(f))=k2fs2G(f)+(k1fs1−k2fs2)H1(f)∗G(f)
G(f)=G(f)*(k_1f_{s1}H_1(f)+k_2f_{s2}H_2(f))\\=G(f)*(k_2f_{s2}\delta(f)+(k_1f_{s1}-k_2f_{s2})H_1(f))\\
=k_2f_{s2}G(f)+(k_1f_{s1}-k_2f_{s2})H_1(f)*G(f)
G(f)=G(f)∗(k1fs1H1(f)+k2fs2H2(f))=G(f)∗(k2fs2δ(f)+(k1fs1−k2fs2)H1(f))=k2fs2G(f)+(k1fs1−k2fs2)H1(f)∗G(f)
可见,只要合理选择fs1f_{s1}fs1和fs2f_{s2}fs2,即可保证H1(f)H_1(f)H1(f)的任意性,且恢复出G(f)G(f)G(f).
令
k2fs2=1k1fs1−k2fs2=0
k_2f_{s2}=1\\
k_1f_{s1}-k_2f_{s2}=0
k2fs2=1k1fs1−k2fs2=0
可得k1=1fs1k_1 = \frac{1}{f_{s1}}k1=fs11,k2=1fs2k_2 = \frac{1}{f_{s2}}k2=fs21.
G(f)=Ws1′(f)fs1+Ws2′(f)fs2
G(f)=\frac{W'_{s1}(f)}{f_{s1}}+\frac{W'_{s2}(f)}{f_{s2}}
G(f)=fs1Ws1′(f)+fs2Ws2′(f)