Question about delayed sampled sinusoid math expression

Question

I have been studying the digital audio processing by using the book <Designing Audio Effect Plugins in C++>.

For analog Sinusoid:

Complex Sinusoid = $e^{jωt}$

Delayed Sinusoid = $e^{jω(t−n)} = e^{jwt} * e^{-jwn}$, a delay of n seconds

For digital sampled version:

sampled complex sinusoid = $e^{jωnT}$, T is interval for each sample, n is the index of sample

I understand all above, but I got confused about the delayed sampled sinusoid which described as: $e^{jω ( nT −M )}$, M = samples of delay

But I think it should be described as $e^{jωT( n − M )}$, since the T is a constant for a fixed sample rate, n and M has the same unit.

At first I thought that maybe a typo, but the following computation parts of the book are all using the $e^{jω ( nT −M )}$ as basis.

Anyone can explain it for me?

Answer 1

Indeed a typo

$e^{j \omega ((n - k)T)}$ would be the delay for $k$ samples, the $M$ in the expression $e^{j \omega(nT - M)}$ is the delay (in seconds if $\omega$ is in $rad/s$)

Answer 2

It seems like an incorrect notation.

When a function of time is sampled, $t = nT_s$ is considered where $n$ is the sample index and $T_s$ is the sampling period.

Generally, if a function of time is given as e.g. $x(t)$, the sampled function is written as $x[n]$ rather than $x(nT_s)$ but this is an acceptable notation too.

In the case of the given exponential function, let’s say $y(t) = e^{j{\omega}t}$, the sampled function will be $y[n] = e^{j{\omega}n}$.

If the sample function $y[n]$ is shifted $M$ samples towards the right-hand side of the sample axis for $M>0$, the resultant shifted function becomes $y[n-M] = e^{j{\omega}[n-M]}$.

Alternatively, if both the time and samples are taken into account at the same time, the shifted function can be written down as $y((n-M)T_s) = e^{j{\omega}(n-M)T_s}$.

However, the shifting notation $(nT_s-{M})$ is wrong as in this case, the shifting is done in the time domain as $nT_s$ is still representing a moment in time because $n$ is a constant value and the sampling period is multiplied by a constant. This process results in a moment in the units of seconds, milliseconds, etc.

This incorrect notation is analogous to subtracting a current value from a voltage value.

I prefer to use the sample function notation e.g. $x[n]$ rather than $x(nT_s)$ to minimise confusion.