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 , 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 and 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 considered to be so large that it is effectively infinite, fractional-delay by can be modeled as convolving the samples with followed by sampling at . In this case, a linear phase term has been introduced in the interpolator frequency response, giving,
Orders 1 to 5 on a fractional delay of 0.4 samples
Linear Interpolation Frequency Response