BAF Winter 2004 Page 1 of 4 Simulating Waveforms in MATLAB By Bruce Ferguson with changes by Mark A. Yoder Simulating waveforms is one operation for which MATLAB has been specifically designed. However, transitioning from the method we use to represent signals in class analysis and using Maple to the method used in MATLAB takes some explanation. In your introductory courses, most of the work has been performed using continuous functions and variables, supported by the continuous mathematics you have learned. However, (digital) computers cannot represent continuous variables. Consider a time domain function x(t). Formally, x(⋅) is a function of an argument. When we write x(t), we have assigned the variable “t” as the input to the function “x”. In this view, x(⋅) is a box containing a machine. We input the argument (a specific value t0) into the box, and a value of the function comes out (x(t0)). A plot of x(t) versus t is a convenient means of representing the function x. The variable “t” is often taken to be continuous, and roughly correlating to the thing we call time. In other words, the variable t can take on any value we would like, to an infinite precision. We typically describe t as being a variable covering a range of values, such as 0<t<10 sec, and are then interested in the values the function x takes using the values of t as input to the function. When the input argument is continuous variable, the function x(t) becomes a continuous function as well. However, the function x does not require a continuous variable as input. If a discrete variable td is applied as input to the function x, the function x(td) becomes discrete. Examples of both situations for the function x(⋅) = cos(⋅) are shown in Figure 1. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-101The function x(t)time (sec)x(t)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-101The simulated function x(t)time td (sec)x(td) Figure 1. Continuous- and discrete-time version of a function x(t).BAF Winter 2004 Page 2 of 4 When working with MATLAB, we must shift our focus to discrete variables. When we describe the discrete variable td, we recognize that the values of td are limited. Most often, we describe td as covering the range 0< td <10, using N points. When N is 100, the variable td takes on the values {0.1, 0.2, 0.3 … 9.9}. In MATLAB, we represent the variable td as a vector containing the various elements of td, i.e. [][]1 2 3 1000.10.2 0.3 0.3 9.9dttttt==LL The variable td in MATLAB is thus a 100 element row vector. The values of each element of the vector are determined from the range to be covered and the number of points used to cover that range. There are two simple ways to create a time vector in MATLAB. The first method uses a built-in function “linspace” to automatically create a vector given a range and number of points. (Type “help linspace” at the MATLAB command prompt to get complete details.) The syntax for using linspace is rather straightforward: t=linspace(begin value, end value, number of points); t=linspace(0.1,9.9,100) It remains to determine what the range of values should be and how many points to use over the desired range. Often, we simulate periodic signals in MATLAB. The fact that the signal is periodic gives us an easy method to specify the range and the number of points: 1. Determine the period of the waveform (period) 2. Choose a convenient starting time (tstart) 3. Choose how many periods of the waveform to simulation (numper, at least 4) 4. The duration of the time vector is then (duration=numper*period) 5. Choose how many points per period to simulate (nsper, at least 10) 6. Calculate the total number of points N=nsper*numper 7. The spacing between points is dt=duration/N In MATLAB syntax, this translates to: period = 1; tstart = 0.1; numper = 4; duration = numper * period; nsper = 16; N = numper * nsper; dt = duration / N; td = linspace(tstart, duration+tstart-dt , N); We have now created a vector td containing N points covering the range ( tstart … (tstart+duration) ). You can verify the number of points in the vector td by using a built-in function for that purpose - “length(td)”.BAF Winter 2004 Page 3 of 4 Another method for creating a time vector uses the colon notation. This method is best if you know the range of time to cover and resolution (the spacing between time values). The syntax is straightforward: td = [tstart, dt, tfinish]; td = [tstart, dt, duration+tstart-dt]; td = [0.1, 0.1, 9.9]; The square brackets are used to define the elements of an array. The resolution (here called dt) will be discussed further below. The function x(td) can easily be produced in MATLAB. Since td is a set of discrete variables, we expect x to become a set of discrete variables. Specifically, in MATLAB, since td is a vector, we would expect x(td) to also be a vector. The assignment statement “x=cos(2*pi/T*td);” is the MATLAB syntax for calculating a vector x containing values of the function x evaluated at the element values of the vector td, i.e. [][]123 12cos ;()()() ( )cos(0.1) cos(0.2) cos(0.3) cos(9.9)dNxtTxt xt xt xtπ−⎛⎞=⎜⎟⎝⎠==LL Note that the argument of the cos function in this case is actually 2πftd. In this case, MATLAB automatically creates x as a vector based on the fact that the argument to the function x, which is td, is a vector. This is called “implicit type assignment”. We can now use the vectors td and x in all sorts of manners, for example to plot the function x(t). The plot function can either display the elements of a vector variable versus array index (“plot(x)”), or displayed versus some other vector (“plot(td,x)”). The latter is the form we normally use to display waveforms. At this point an important difference in the way students perceive the operation of Maple and MATLAB has been revealed. In Maple, when we wish to plot cos(t), we use the plot command “plot cos(t), t=1..10”. This automatically creates a vector t and computes a vector cos(t) and then plots the vector cos(t) versus the vector t. We just don’t see the vector operations. In MATLAB, we explicitly perform the vector operations by creating the vector td and then computing the vector cos(td). We might also wish to create a new function y=x2 (using “y=x.*x”). Since x is a vector, which just a row matrix, we must be mindful of which type of operation we wish to perform when
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