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Introduction

Hey it's a me again @drifter1!

Today we continue with my mathematics series about Signals and Systems in order to cover Discrete-Time Modulation.

So, without further ado, let's dive straight into it!

Modulation in Discrete-Time

After covering continuous-time modulation, its now time for discrete-time modulation. Mulation is again a property of the Fourier Transform. Its the Fourier Transform of the multiplication/product of two signals, which turns into a convolution of their individual Fourier Transforms.

Consider x[n] to be the discrete-time signal to be "modulated". The signal c[n] which is used to modulate x[n] is known as the carrier signal. As a result, the modulation property can be written as:

The discrete-time Fourier Transform is a periodic function of frequency, which in turn makes the convolution also periodic. This is also the main difference with the continuous-time FT, where the convolution is linear.

Applications of Discrete-Time Modulation

The mathematics might seem similar, but the applications are different. As we saw last time, continuous-time modulation is a major concept of communication systems, as it is used for transmitting signals over various types of channels. But, in modern communication systems, these signals might have to go through a lot of stages and types of modulation. Thus, this conversion from modulation to another is better implemented digitally, with digital meaning discrete-time signals.

Discrete-time modulation is also useful in Filtering as it can be used to implement filters with a variable center frequency. Although it might be simpler to implement a fixed filter (in either continous- or discrete-time) and then use modulation to shift the signal spectrum in relation to the fixed filter center frequency.

Sinusoidal and Complex Expnential Carriers

Similar to continuous-time, the carrier signals in discrete-time can also be complex exponentials or sinusoidals. The Fourier transform of an complex exponential signal is:

whilst the sinusoidal signal has a FT of:

A sinusoidal signal can easily be defined with the use of complex exponentials, as follows:

which also explains why the two graphical representations are so similar.

Pulse Carrier

Due to the nature of discrete samples, another useful type of carrier is also a periodic pulse, which is constant for some samples of the period and zero for the remainder. This signal can then be used in modulation to extract time slices of the modulating signal. Visually, pulse carriers look as follows:

The pulse train with the smallest time of being "on" can easily correspond to a periodic train of impulses, or even unit samples. A impulse train carrier allows sampling the modulating input signal, and leads to another very important concept, known as the sampling theorem. We will get into sampling in the next part of the series, but let's just note that it provides a major bridge between continuous-time and discrete-time systems and signals.

Pulse carriers can be useful in both continuous-time and discrete-time applications. In continuous-time amplitude modulation with a pulse carrier, this leads to interesting results, such as:

Using this pulse carrier only specific time slices of the original signal x(t) are allowed to pass to the output y(t).

Modulation and Demodulation

Similar to continuous-time, when using sinusoidal carriers demodulation is defined as a modulation with a sinusoidal followed by a low-pass filter.

Complex exponentials just require a re-modulation in order to recover the original signal.

Let's also mention that modulating a signal with the signal (-1)ⁿ, passing it through a low-pass filter and re-modulating it again, is an easy way of implementing a high-pass filter:

Time-Division Multiplexing

When representing the data for transmission, using pulse trains allows a specific type of multiplexing known as time-division multiplexing. Using TDM, the time slices from signals in other channels can be inserted to the carrier as well. Recovery or demodulation is also quite simple.

Visually, using amplitude modulation with a pulse carrier, multiple signals (and so multiple data) can be multiplexed through TDM, as follows:

RESOURCES:

References

Alan Oppenheim. RES.6-007 Signals and Systems. Spring 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, License: Creative Commons BY-NC-SA.

Images

https://commons.wikimedia.org/wiki/File:From_Continuous_To_Discrete_Fourier_Transform.gif

Mathematical equations used in this article were made using quicklatex.

Block diagrams and other visualizations were made using draw.io and GeoGebra

Previous articles of the series

Basics

LTI Systems and Convolution

Fourier Series and Transform

Filtering, Sampling, Modulation, Interpolation

Filtering → Convolution Property, Ideal Filters, Series R-C Circuit and Moving Average Filter Approximations
Continuous-Time Modulation → Getting into Modulation, AM and FM, Demodulation

Final words | Next up

And this is actually it for today's post!

Next time we will get into Sampling...

See Ya!

Keep on drifting!

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