Lecture 04 - Cross-Domain Processing

Signals can be processed in both the continuous time (CT) and discrete time (DT) domains. Oftentimes processing in one domain is more practical than the other, so some means of conversion between them is required.

In this lecture we will discuss the ways we model so-called “equivalent” systems and why they are so essential to communcations engineering.

Primitives

Before formalizing the relationship between continuous and discrete time systems it is important to reinforce several concepts that are necessary for our derivations. Nothing here should be new.

1. Modulation by an Impulse Train

A signal $x(t)$ can be modulated by a train of impulses spaced by $T_s$. The resulting signal $y(t)$ is still continuous time despite its resemblance to a discrete time signal.

\[ \begin{aligned} y(t) = \sum_{n=-\infty}^{\infty}x(nT_s) \, \delta(t-nT_s) \; &\longleftrightarrow \; \mathbb{Y} (j \omega) = \frac{1}{T_s} \sum_{k=-\infty}^{\infty} \mathbb{X} (j (\omega - \tfrac{k2\pi}{T_s})) \\ \; &\longleftrightarrow \; \mathbb{Y} (f) = \frac{1}{T_s} \sum_{k=-\infty}^{\infty} \mathbb{X} (f - \tfrac{k}{T_s}) \\ \end{aligned} \]

2. DT Sequence to CT Impulse Train

A discrete time sequence $x[n]$ can be converted to continuous time by multiplication with an impulse train.

\[ \begin{aligned} y(t) = \sum_{n=-\infty}^{\infty}x[n] \, \delta(t-nT_s) \; \; \; &\longleftrightarrow \; \; \; \mathbb{Y} (j \omega) = \mathbb{X} (e^{j \omega T_s}) \\ \; \; \; &\longleftrightarrow \; \; \; \mathbb{X} (e^{j \Omega}) = \mathbb{Y} (j \tfrac{\Omega}{T_s}) \\ \end{aligned} \]

3. Sampling

A continuous time signal can be sampled at some interval to produce a discrete time sequence.

\[ \begin{aligned} x[n] = y(nT_s) \; \; \; &\longleftrightarrow \; \; \; \mathbb{X} (e^{j \Omega}) = \frac{1}{T_s} \sum_{k=-\infty}^{\infty} \mathbb{Y} (j (\tfrac{\Omega - k2\pi}{T_s})) \\ \; \; \; &\longleftrightarrow \; \; \; \mathbb{X} (e^{2 \pi u}) = \frac{1}{T_s} \sum_{k=-\infty}^{\infty} \mathbb{Y} (\tfrac{u - k}{T_s}) \\ \end{aligned} \]

4. Interpolation with an LTI Filter $h(t)$

A discrete time sequence can be converted back to a continuous time signal by application of a sinc-interpolation filter.

\[ z(t) = \sum_{n=-\infty}^{\infty}x[n] \, h(t-nT_s) \; \; \; \longleftrightarrow \; \; \; \mathbb{Z} (j \omega) = \mathbb{H} (j \omega) \, \mathbb{X} (e^{j \omega T_s}) \]

Note that the resulting signal $z(t)$ is a sum of the input signal $x[n]$ times time-delayed versions of the filter $h(t)$.

Continuous Time to Discrete Time (and back)

We can convert a continuous time signal $x(t)$ to a discrete time signal $x[n]$ and then back to a continuous time signal $\tilde{x} (t)$ with the following construction:

Recall from the sampling theorem that if $x(t)$ is bandlimited to a maximum frequency $\omega_0$ and $\frac{2\pi}{T_s} > 2 \omega_0$, then $\tilde{x} (t) = x(t)$.

Note: $x(t)$ and $\tilde{x} (t)$ are equal in the sense that $\int_{-\infty}^{+\infty} | \tilde{x} (t) - x(t) |^2dt = 0$ - i.e., the energy of the error is $0$. This is not strictly the same as saying that - $x(t) = \tilde{x} (t)$ for all $t$.

Discrete Time Processing of Continuous Time Signals

One important benefit of the above property is that we can do discrete time processing of continuous time signals. The block diagram below demonstrates such a system, where $g[n]$ is some discrete time filter used to process the signal.

This filter could be anything from a basic low-pass filter to a video codec - the idea is that we can use discrete time techniques (computers) to process continous time signals.

We can combine concepts from sampling (3), bandlimited interpolation (4), and the Fourier Transform property $\mathbb{Y} (e^{j \Omega}) = \mathbb{G} (e^{j \Omega}) \mathbb{X} (e^{j \Omega})$ to find that the output Fourier transform is the scaled version of the filter’s frequency response multiplied by the input:

\[ \mathbb{Y} (j \omega) = \begin{cases} \mathbb{G} (e^{j \omega T_s}) \mathbb{X} (j \omega) & | \omega | < \frac{\pi}{T_s} \\ 0 & \text{otherwise} \end{cases} \]

Considering the equation above, we see that this relation only holds for a limited range of frequencies $\omega$. This limitation is due to the cutoff frequency $\frac{\pi}{T_s}$ of the ideal low-pass filter in the bandlimited interpolation stage. Keep in mind that the discrete time Fourier transform of the filter response $\mathbb{G}(e^{j \omega T_s})$ is periodic in frequency. The sampled signal $x[n]$ has the same property, with the same periodicity. The bandlimited interpolation selects a single image of both $\mathbb{G}(e^{j \omega T_s})$ and $x[n]$, namely the image centered on the origin of the frequency axis.

Considering this and looking at the figure, we see that the entire system (in the dashed-line box) behaves like an equivalent CT LTI system with

\[ \mathbb{G}_{eq} (j \omega) = \begin{cases} \mathbb{G} (e^{j \omega T_s}) & | \omega | < \frac{\pi}{T_s} \\ 0 & \text{otherwise} \end{cases} \]

Despite the fact that the processing inside the box is in discrete time, the system as a whole is truly a continuous time system because it has a continuous time input and a continuous time output. This allows, in general, for the implementation of discrete time processing steps within a continuous time system.

Note: if $x(t)$ is not bandlimited or aliasing occurs, the relationship is more involved and the equivalent CT system is not LTI.

Discrete Time to Continuous Time (and back)

A similar process exists for the reverse situation, in which a discrete time signal is converted to continuous time and back. We can combine concepts from sampling (3) and bandlimited interpolation (4) to get

\[ \tilde{x} [n] = x[n], \; n \in \mathbb{Z} \]

Notice that $x(t)$ must always be bandlimited with max frequency less than $\frac{\pi}{T_s}$ so there can be no aliasing.

Continuous Time Processing of Discrete Time Signals

The conversion from discrete time to continuous time and back is useful when we wish to perform continuous time processing within a discrete time system. This can be modeled as the insertion of some CT LTI system represented by $g(t)$ between the interpolation and sampling stages, as shown above.

Refer to the block diagram above. By combining concepts from sampling (3), bandlimited interpolation (4), and the property of the Fourier Transform $\mathbb{Y} (j \omega) = \mathbb{G} (j \omega) \mathbb{X} (j \omega)$ we can write an expression for the output DT Fourier transform of this system:

\[ \mathbb{Y} (e^{j \Omega}) = \mathbb{G} (j \tfrac{\Omega}{T_s}) \, \mathbb{X} (e^{j \Omega}) \; \; \; \text{for} \; | \Omega | < \pi \]

Here, $\mathbb{G}(j \frac{\Omega}{T_s})$ is the scaled version of the CT filter’s frequency response, and $\mathbb{X}(e^{j \Omega})$ is the DT Fourier transform of the input signal $x[n]$. Note that $\mathbb{Y} (e^{j \Omega})$ is periodic with period $2\pi$, i.e., the entire system behaves like an equivalent DT LTI system with

$$ {eq} (e^{j }) = {k=-}^ (j())

(j ) = \[\begin{cases} \mathbb{G} (j \omega) & | \omega | < \frac{\pi}{T_s} \\ 0 & \text{otherwise} \end{cases}\]

Here, $\mathbb{G}(j \omega)$ is the CT Fourier transform of $g(t)$. Since $x(t)$ is bandlimited by the interpolation filter, only the components of $\mathbb{G}(j \omega)$ which fall within that filter’s passband are retained in the multiplication. This bandlimited response $\tilde{\mathbb{G}}(j \omega)$ is then made periodic in frequency, in order to satisfy the DT output requirements of the overall system, and called $\mathbb{G}_{eq}(e^{j \Omega})$. In discrete time, Fourier transforms must be periodic. We can now rewrite the frequency domain input-output relationship as:

\[ \mathbb{Y}(e^{j \omega}) = \mathbb{G}_{eq} (e^{j \omega}) \mathbb{X}(e^{j \omega}) \]

In the time domain, the corresponding relationship is:

\[ g_{eq}[n] = \left(g(t) * \mathrm{sinc}(t / T_s) \right) \big\rvert_{t=nT_s} \]

The system within the dashed-line box is equivalently a discrete time LTI system, as long as $g(t)$ is an LTI system. This equivalence is a very important concept when considering communications systems: discrete time signals can be transmitted using continuous time propagation modes, and recovered (sampled) again as discrete time signals. By modeling this as a DT LTI signal processing system, we can suppress the details of the physical propagation mode and focus on the details of the DT system design which are applicable when using any such mode of continuous time propagation (i.e., wired or wireless transmission links).

NOTES FOR REVIEWERS:

still would like to include some examples in Python to help students get familiar with modelling these systems. Perhaps figures for the primitives would be helpful?
need to brainstorm on exercises to include so that students can get some practice with math - particularly with regard to converting from CT to DT and how that imparts periodicity
surely some information or explanations have been missed here, please address any such issue