Linearly interpolated fractional delay is equivalent to filtering and resampling a weighted impulse train (the input signal samples) with a continuous-time filter having the simple triangular impulse response
Convolution of the weighted impulse train with produces a continuous-time linearly interpolated signal
This continuous result can then be resampled at the desired fractional delay.
In discrete time processing, the operation Eq.(4.5) can be approximated arbitrarily closely by digital upsampling by a large integer factor , delaying by samples (an integer), then finally downsampling by , as depicted in Fig.4.7 . The integers and are chosen so that , where the desired fractional delay.
Since linear interpolation can be expressed as a convolution of the samples with a triangular pulse, we can derive the frequency response of linear interpolation. Figure 4.7 indicates that the triangular pulse serves as an anti-aliasing lowpass filter for the subsequent downsampling by . Therefore, it should ideally ``cut off'' all frequencies higher than .
The one-sample rectangular pulse is shown in Fig.4.8 and may be defined analytically as
Since linear interpolation is a convolution of the samples with a triangular pulse (from Eq.(4.5)), the frequency response of the interpolation is given by the Fourier transform , which yields a sinc function. This frequency response applies to linear interpolation from discrete time to continuous time. If the output of the interpolator is also sampled, this can be modeled by sampling the continuous-time interpolation result in Eq.(4.5), thereby aliasing the sinc frequency response, as shown in Fig.4.9.
In slightly more detail, from , and sinc, we have
The Fourier transform of is the same function aliased on a block of size Hz. Both and its alias are plotted in Fig.4.9. The example in this figure pertains to an output sampling rate which is times that of the input signal. In other words, the input signal is upsampled by a factor of using linear interpolation. The ``main lobe'' of the interpolation frequency response contains the original signal bandwidth; note how it is attenuated near half the original sampling rate ( in Fig.4.9). The ``sidelobes'' of the frequency response contain attenuated copies of the original signal bandwidth (see the DFT stretch theorem), and thus constitute spectral imaging distortion in the final output (sometimes also referred to as a kind of ``aliasing,'' but, for clarity, that term will not be used for imaging distortion in this book). We see that the frequency response of linear interpolation is less than ideal in two ways:
- The spectrum is ``rolled'' off near half the sampling rate. In fact, it is nowhere flat within the ``passband'' (-1 to 1 in Fig.4.9).
- Spectral imaging distortion is suppressed by only 26 dB (the level of the first sidelobe in Fig.4.9.
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,
Large Delay Changes
First-Order Allpass Interpolation