## A single time-delay can also be expressed in the frequency-domain.

Another way to state, that a stream of time-domain samples has been given a time-delay, is simply to state that each frequency-coefficient has been given a phase-shift, that depends both on the frequency of the coefficient, and on the intended time-delay.

A concern that some readers might have with this, is the fact that a number of samples need to be stored, in order for a time-delay to be executed in the time-domain. But as soon as differing values for coefficients, for a Fourier Transform, are spaced closer together, indicating in this case a longer time-delay, its computation also requires that a longer interval of samples in the time-domain need to be combined.

Now, if the reader would like to visualize what this would look like, as a homology to a graphical equalizer, then he would need to imagine a graphical equalizer the sliders of which can be made negative – i.e. one that can command, that one frequency come out inverted – so that then, if he was to set his sliders into the accurate shape of a sine-wave that goes both positive and negative in its settings, he should obtain a simple time-delay.

But there is one more reason for which this homology would be flawed. The type of Fourier Transform which is best-suited for this, would be the Discrete Fourier Transform, not one of the Discrete Cosine Transforms. The reason is the fact that the DFT accepts complex numbers as its terms. And so the reader would also have to imagine, that his equalizer not only have sliders that move up and down, but sliders with little wheels on them, from which he can give a phase-shift to one frequency, without changing its amplitude. Obviously graphical equalizers for music are not made that way.

## About the Amplitudes of a Discrete Differential

One of the concepts which exist in digital signal processing, is that the difference between two consecutive input samples (in the time-domain) can simply be output, thus resulting in a differential of some sort, even though the samples of data do not represent a continuous function. There is a fact which must be observed to occur at (F = N / 2) – i.e. when the frequency is half the Nyquist Frequency, of (h / 2) , if (h) is the sampling frequency.

The input signal could be aligned with the samples, to give a sequence of [s0 … s3] equal to

0, +1, 0, -1

This set of (s) is equivalent to a sine-wave at (F = N / 2) . Its discrete differentiation [h0 … h3] would be

+1, +1, -1, -1

At first glance we might think, that this output stream has the same amplitude as the input stream. But the problem becomes that the output stream is by same token, not aligned with the samples. There is an implicit peak in amplitudes between (h0) and (h1) which is greater than (+1) , and an implicit peak between (h2) and (h3) more negative than (-1) . Any adequate filtering of this stream, belonging to a D/A conversion, will reproduce a sine-wave with a peak amplitude greater than (1).

(Edit 03/23/2017 : )

In this case we can see, that samples h0 and h1 of the output stream, would be phase-shifted 45⁰ with respect to the zero crossings and to the peak amplitude, that would exist exactly between h0 and h1. Therefore, the amplitude of h0 and h1 will be the sine-function of 45⁰ with respect to this peak value, and the actual peak would be (the square root of 2) times the values of h0 and h1.

(Erratum 11/28/2017 —

And so a logical question which anybody might want an answer to would be, ‘Below what frequency does the gain cross unity gain?’ And the answer to that question is, somewhat obscurely, at (N/3) . This is a darned low frequency in practice. If the sampling rate was 44.1kHz, this is achieved somewhere around 7 kHz, and music, for which that sampling rate was devised, easily contains sound energy above that frequency.

Hence the sequences which result would be:

s = [ +1, +1/2, -1/2, -1, -1/2, +1/2 ]

h = [ +1/2, -1/2, -1, -1/2, +1/2, +1 ]

What follows is also a reason for which by itself, DPCM offers poor performance in compressing signals. It usually needs to be combined with other methods of data-reduction, thus possibly resulting in the lossy ADPCM. And another approach which uses ADPCM, is aptX, the last of which is a proprietary codec, which minimizes the loss of quality that might otherwise stem from using ADPCM.

I believe this observation is also relevant to This Earlier Posting of mine, which implied a High-Pass Filter with a cutoff frequency of 500 Hz, that would be part of a Band-Pass Filter. My goal was to obtain a gain of at most 0.5 , over the entire interval, and to simplify the Math.

— End of Erratum. )

(Posting shortened here on 11/28/2017 . )

Dirk