A discrete-time system is anything that takes a discrete-time signal as input and generates a discrete-time signal as output.1 The concept of a system is very general. It may be used to model the response of an audio equalizer or the performance of the US economy.

In electrical engineering, continuous-time signals are usually processed by electrical circuits described by differential equations. For example, any circuit of resistors, capacitors and inductors can be analyzed using mesh analysis to yield a system of differential equations. The voltages and currents in the circuit may then be computed by solving the equations.

The processing of discrete-time signals is performed by discrete-time systems. Similar to the continuous-time case, we may represent a discrete-time system either by a set of difference equations or by a block diagram of its implementation. For example, consider the following difference equation.

This equation represents a discrete-time system. It operates on the input signal x(n)x(n) to produce the output signal y(n)y(n). This system may also be defined by a system diagram as in Figure 1.

Figure 1: Diagram of a discrete-time system. (D = unit delay)

Mathematically, we use the notation y=S[x]y=S[x] to denote a discrete-time system SS with input signal x(n)x(n) and output signal y(n)y(n). Notice that the input and output to the system are the complete signals for all time n. This is important since the output at a particular time can be a function of past, present and future values of x(n)x(n).

It is usually quite straightforward to write a computer program to implement a discrete-time system from its difference equation. In fact, programmable computers are one of the easiest and most cost effective ways of implementing discrete-time systems.

While equation Equation 1 is an example of a linear time-invariant system, other discrete-time systems may be nonlinear and/or time varying. In order to understand discrete-time systems, it is important to first understand their classification into categories of linear/nonlinear, time-invariant/time-varying, causal/noncausal, memoryless/with-memory, and stable/unstable. Then it is possible to study the properties of restricted classes of systems, such as discrete-time systems which are linear, time-invariant and stable.

Write two Matlab functions that will apply the differentiator and integrator systems, designed in the "Example Discrete-time Systems" section, to arbitrary input signals. Then apply the differentiator and integrator to the following two signals for -10≤n≤20-10≤n≤20.

Hint: To compute the function u(n)u(n) for -10≤n≤20-10≤n≤20, first set n = -10:20, and then use the Boolean expression u = (n>=0).

For each of the four cases, use the subplot and stem commands to plot each input and output signal on a single figure.

In this section, we will study the effect of two discrete-time filters. The first filter, y=S1[x]y=S1[x], obeys the difference equation

and the second filter, y=S2[x]y=S2[x], obeys the difference equation

Write Matlab functions to implement each of these filters.

Now use these functions to calculate the impulse response of each of the following 5 systems: S1S1, S2S2, S1(S2)S1(S2) (i.e., the series connection with S1S1 following S2S2), S2(S1)S2(S1) (i.e., the series connection with S2S2 following S1S1), and S1+S2S1+S2.

For this section download the music.au file. For help on how to play audio signals click here.

Use the command auread to load the file music.au into Matlab. Then use the Matlab function sound to listen to the signal.

Next filter the audio signal with each of the two systems S1S1 and S2S2 from the previous section. Listen to the two filtered signals.

Consider the system y=S2[x]y=S2[x] from the "Difference Equations" section. Find a difference equation for a new system y=S3[x]y=S3[x] such that δ=S3[S2[δ]]δ=S3[S2[δ]] where δδ denotes the discrete-time impulse function δ(n)δ(n). Since both systems S2S2 and S3S3 are LTI, the time-invariance and superposition properties can be used to obtain x=S3[S2[x]]x=S3[S2[x]] for any discrete-time signal xx. We say that the systems S3S3 and S2S2 are inverse filters because they cancel out the effects of each other.

Hint: The system y=S3[x]y=S3[x] can be described by the difference equation

where aa and bb are constants.

Write a Matlab function y = S3(x) which implements the system S3S3. Then obtain the impulse response of both S3S3 and S3[S2[δ]]S3[S2[δ]].

For this section download the zip file bbox.zip.

Often it is necessary to determine if a system is linear and/or time-invariant. If the inner workings of a system are not known, this task is impossible because the linearity and time-invariance properties must hold true for all possible inputs signals. However, it is possible to show that a system is non-linear or time-varying because only a single instance must be found where the properties are violated.

The zip file bbox.zip contains three "black-box" systems in the files bbox1.p, bbox2.p, and bbox3.p. These files work as Matlab functions, with the syntax y=bboxN(x), where x and y are the input and the output signals, and N = 1, 2 or 3. Exactly one of these systems is non-linear, and exactly one of them is time-varying. Your task is to find the non-linear system and the time-varying system.

For this section download stockrates.mat. For help on loading Matlab files click here.

Load stockrates.mat into Matlab. This file contains a vector, called rate, of daily stock market exchange rates for a publicly traded stock.

Apply filters Equation 4 and Equation 5 from the "Stock Market Example" section of the background exercises to smooth the stock values. When you apply the filter of Equation 4 you will need to initialize the value of avgvalue(yesterday)avgvalue(yesterday). Use an initial value of 0. Similarly, in Equation 5, set the initial values of the "value" vector to 0 (for the days prior to the start of data collection). Use the subplot command to plot the original stock values, the result of filtering with Equation 4, and the result of filtering with Equation 5.