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Interpolation

Module by: Phil Schniter

Summary: Introduction of interpolation and it's application.

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Interpolation

Interpolation is the process of upsampling and filtering a signal to increase its effective sampling rate. To be more specific, say that xm x m is an (unaliased) TT-sampled version of x c t x c t and vn v n is an LL-upsampled version version of xm x m . If we filter vn v n with an ideal πLL-bandwidth lowpass filter (with DC gain LL) to obtain yn y n , then yn y n will be a TL TL -sampled version of x c t x c t . This process is illustrated in Figure 1.

Figure 1
Figure 1 (m10444fig1.png)

We justify our claims about interpolation using frequency-domain arguments. From the sampling theorem, we know that TT- sampling x c t x c t to create xn x n yields

Xⅇⅈω=1T∑k X c ⅈω−2πkT X ω 1 T k k X c ω 2 k T (1)
After upsampling by factor LL, Equation 1 implies Vⅇⅈω=1T∑k X c ⅈωL−2πkT=1T∑k X c ⅈω−2πLkTL V ω 1 T k k X c ω L 2 k T 1 T k k X c ω 2 L k T L Lowpass filtering with cutoff πL L and gain LL yields Yⅇⅈω=LT∑kL∈ℤ X c ⅈω−2πLkTL=LT∑l X c ⅈω−2πlTL Y ω L T kL k L X c ω 2 L k T L L T l l X c ω 2 l T L since the spectral copies with indices other than k=lL klL (for l∈ℤ l) are removed. Clearly, this process yields a TL TL-shaped version of x c t x c t . Figure 2 illustrates these frequency-domain arguments for L=2L2.

Figure 2
Figure 2 (m10444fig2.png)

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