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Special Cases

In the limiting case of $ L=1$, the input and output sampling rates are equal, and all sidelobes of the frequency response $ H_l(f)$ (partially shown in Fig.4.9) alias into the main lobe.

If the output is sampled at the same exact time instants as the input signal, the input and output are identical. In terms of the aliasing picture of the previous section, the frequency response aliases to a perfect flat response over $ fT\in[-1,1]$, with all spectral images combining coherently under the flat gain. It is important in this reconstruction that, while the frequency response of the underlying continuous interpolating filter is aliased by sampling, the signal spectrum is only imaged--not aliased; this is true for all positive integers $ L$ and $ M$ in Fig.4.7.

More typically, when linear interpolation is used to provide fractional delay, identity is not obtained. Referring again to Fig.4.7, with $ L$ considered to be so large that it is effectively infinite, fractional-delay by $ \tau<1$ can be modeled as convolving the samples $ x(n)$ with $ h_l(t-\tau)$ followed by sampling at $ t=nT$. In this case, a linear phase term has been introduced in the interpolator frequency response, giving,

$\displaystyle H_\tau(f) \isdef e^{-j\tau 2\pi f} H_l(f) $

so that now downsampling to the original sampling rate does not yield a perfectly flat amplitude response for $ fT\in[-1,1]$ (when $ \tau $ is non-integer). Moreover, the phase response is nonlinear as well; a sampled symmetric triangular pulse is only linear phase when the samples fall symmetrically about the midpoint. Some example frequency-responses for various delays $ \tau $ are graphed in Fig.4.2.

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Orders 1 to 5 on a fractional delay of 0.4 samples
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Linear Interpolation Frequency Response