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 The function x t x t 1 0 1 0 0 5 1 1 5 2 2 5 time sec 3 3 5 4 4 5 3 5 4 4 5 The simulated function x t x td 1 0 1 0 0 5 1 1 5 2 2 5 time td sec 3 Figure 1 Continuous and discrete time version of a function x t BAF Winter 2004 Page 1 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 td t1 t2 t3 L t100 0 1 0 2 0 3 0 3 L 9 9 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 2 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 2 x cos td T x t1 x t2 x t3 L x t N 1 cos 0 1 cos 0 2 cos 0 3 L cos 9 9 Note that the argument of the cos function in this case is actually 2 ftd 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 using MATLAB The assignment y …
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