The purpose of this lab is to illustrate the properties of continuous and discrete-time signals using digital computers and the Matlab software environment. A continuous-time signal takes on a value at every point in time, whereas a discrete-time signal is only defined at integer values of the “time” variable. However, while discrete-time signals can be easily stored and processed on a computer, it is impossible to store the values of a continuous-time signal for all points along a segment of the real line. In later labs, we will see that digital computers are actually restricted to the storage of quantized discrete-time signals. Such signals are appropriately known as digital signals.

How then do we process continuous-time signals? In this lab, we will show that continuous-time signals may be processed by first approximating them by discrete-time signals using a process known as sampling. We will see that proper selection of the spacing between samples is crucial for an efficient and accurate approximation of a continuous-time signal. Excessively close spacing will lead to too much data, whereas excessively distant spacing will lead to a poor approximation of the continuous-time signal. Sampling will be an important topic in future labs, but for now we will use sampling to approximately compute some simple attributes of both real and synthetic signals.

Practically all lab tasks in the ECE438 lab will be performed using Matlab. Matlab (MATrix LABoratory) is a technical computing environment for numerical analysis, matrix computation, signal processing, and graphics. In this section, we will review some of its basic functions. For a short tutorial and some Matlab examples click here.

The "Introduction" mentioned the important issue of representing continuous-time signals on a computer. In the following sections, we will illustrate the process of sampling, and demonstrate the importance of the sampling interval to the precision of numerical computations.

n = 0:2:60;
y = sin(n/6);
subplot(3,1,1)
stem(n,y)

n1 = 0:2:60;
z = sin(n1/6);
subplot(3,1,2)
plot(n1,z)
n2 = 0:10:60;
w = sin(n2/6);
subplot(3,1,3)
plot(n2,w)

Numerical Computation of Continuous-Time Signals

For help on the following topics, click the corresponding link: MatLab Scripts, MatLab Functions, and the Subplot Command.

Background on Numerical Integration

One common calculation on continuous-time signals is integration. Figure 1 illustrates a method used for computing the widely used Riemann integral. The Riemann integral approximates the area under a curve by breaking the region into many rectangles and summing their areas. Each rectangle is chosen to have the same width ΔtΔt, and the height of each rectangle is the value of the function at the start of the rectangle's interval.

Figure 1: Illustration of the Riemann integral

To see the effects of using a different number of points to represent a continuous-time signal, write a Matlab function for numerically computing the integral of the function s i n2 ( 5 t ) s i n2 ( 5 t ) over the interval [ 0 , 2 π ] [ 0 , 2 π ] . The syntax of the function should be I=integ1(N) where I I is the result and N N is the number of rectangles used to approximate the integral. This function should use the sum command and it should not contain any for loops!

Note:

Since Matlab is an interpreted language, for loops are relatively slow. Therefore, we will avoid using loops whenever possible.

Next write an m-file script that evaluates I(N)I(N) for 1≤N≤1001≤N≤100, stores the result in a vector and plots the resulting vector as a function of NN. This m-file script may contain for loops.

Repeat this procedure for a second function J=integ2(N) which numerically computes the integral of exp(t)exp(t) on the interval [0,1][0,1].

INLAB REPORT:

Submit plots of I(N)I(N) and J(N)J(N) versus NN. Use the subplot command to put both plots on a single sheet of paper. Also submit your Matlab code for each function. Compare your results to the analytical solutions from the "Analytical Calculation" section. Explain why I(5)=I(10)=0I(5)=I(10)=0.

For this section download the speech.au file. For instructions on how to load and play audio signals click here.

Digital signal processing is widely used in speech processing for applications ranging from speech compression and transmission, to speech recognition and speaker identification. This exercise will introduce the process of reading and manipulating a speech signal.

First download the speech audio file speech.au, and then do the following:

For this section download the signal1.p function.

In this section you will practice writing .m-files to calculate the basic attributes of continuous-time signals. Download the function signal1.p. This is a pre-parsed pseudo-code file (P-file), which is a “pre-compiled” form of the Matlab function signal1.m. To evaluate this function, simply type y = signal1(t) where t is a vector containing values of time. Note that this Matlab function is valid for any real-valued time, t , so y = signal1(t) yields samples of a continuous-time function.

First plot the function using the plot command. Experiment with different values for the sampling period and the starting and ending times, and choose values that yield an accurate representation of the signal. Be sure to show the corresponding times in your plot using a command similar to plot(t,y).

Next write individual Matlab functions to compute the minimum, maximum, and approximate energy of this particular signal. Each of these functions should just accept an input vector of times, t, and should call signal1(t) within the body of the function. You may use the built-in Matlab functions min and max. Again, you will need to experiment with the sampling period, and the starting and ending times so that your computations of the min, max, and energy are accurate.

Remember the definition of the energy is

Plot the following two continuous-time functions over the specified intervals. Write separate script files if you prefer. Use the subplot command to put both plots in a single figure, and be sure to label the time axes.

Write an .m-script file to stem the following discrete-time function for a=0.8a=0.8, a=1.0a=1.0 and a=1.5a=1.5. Use the subplot command to put all three plots in a single figure. Issue the command orient('tall') just prior to printing to prevent crowding of the subplots.

Repeat this procedure for the function

The word sampling refers to the conversion of a continuous-time signal into a discrete-time signal. The signal is converted by taking its value, or sample, at uniformly spaced points in time. The time between two consecutive samples is called the sampling period. For example, a sampling period of 0.1 seconds implies that the value of the signal is stored every 0.1 seconds.

Consider the signal f(t)=sin(2πt)f(t)=sin(2πt). We may form a discrete-time signal, x(n)x(n), by sampling this signal with a period of TsTs. In this case,

Use the stem command to plot the function f(Tsn)f(Tsn) defined above for the following values of TsTs and nn. Use the subplot command to put all the plots in a single figure, and scale the plots properly with the axis command.

For help on the Matlab random function, click here.

The objective of this section is to show how two signals that “look” similar can be distinguished by computing their average over a large interval. This type of technique is used in signal demodulators to distinguish between the digits “1” and “0”.

Generate two discrete-time signals called “sig1” and “sig2” of length 1,000. The samples of “sig1” should be independent, Gaussian random variables with mean 0 and variance 1. The samples of “sig2” should be independent, Gaussian random variables with mean 0.2 and variance 1. Use the Matlab command random or randn to generate these signals, and then plot them on a single figure using the subplot command. (Recall that an alternative name for a Gaussian random variable is a normal random variable.)

Next form a new signal “ave1(n)” of length 1,000 such that “ave1(n)” is the average of the vector “sig1(1:n)” (the expression sig1(1:n) returns a vector containing the first n elements of “sig1”). Similarly, compute “ave2(n)” as the average of “sig2(1:n)”. Plot the signals “ave1(n)” and “ave2(n)” versus “n” on a single plot. Refer to help on the Matlab plot command for information on plotting multiple signals.

For help on the following topics, click the corresponding link: Meshgrid Command, Mesh Command, and Displaying Images.

So far we have only considered 1-D signals such as speech signals. However, 2-D signals are also very important in digital signal processing. For example, the elevation at each point on a map, or the color at each point on a photograph are examples of important 2-D signals. As in the 1-D case, we may distinguish between continuous-space and discrete-space signals. However in this section, we will restrict attention to discrete-space 2-D signals.

When working with 2-D signals, we may choose to visualize them as images or as 2-D surfaces in a 3-D space. To demonstrate the differences between these two approaches, we will use two different display techniques in Matlab. Do the following: