SIGNALS AND LINEAR SYSTEMS LABORATORY EELE Experiment (2) Introduction to MATLAB - Part (2) Prepared by:

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1 The Islamic University of Gaza Faculty of Engineering Electrical Engineering Department SIGNALS AND LINEAR SYSTEMS LABORATORY EELE 110 Experiment () Introduction to MATLAB - Part () Prepared by: Eng. Mohammed S. Abuwarda Eng. Heba M. Ghurab Fall Semester - 017

2 Experiment () Introduction to MATLAB II 1. Scalar Functions Certain MATLAB functions operate essentially on scalars, but operate element wise when applied to a matrix. Some of these functions are: abs, sign round, floor, ceil sin, cos, tan asin, acos, atan sqrt, exp log (natural log), log10 (log base 10) The following table describes some of the scalar functions: FUNCTIONS DESCRIPTION ABS Abs (x) is the absolute value of the elements of x. SIGN Sign (x) returns 1 if the element x is greater than zero, 0 if it equals zero and 1 if it is less than zero. ROUND Round (x) rounds the elements of x to the nearest integers. SIN Sin (x) is the sine of the elements of x, where x in radian. ASIN asin Inverse sine. EXP exp(x) is the exponential of the elements of x. LOG log(x) is the natural logarithm of the elements of x. LOG10 log10(x) is the base 10 logarithm of the elements of x. Examples: > abs(-) > round(0.5) > 1 > round(0.4) > 0 > floor(4.4) > 4 > ceil(.1) 4 > sin(0.5)

3 .Vector, Matrix and Array Commands Some of MATLAB functions operate essentially on a vector (row or column), and others on an m-by-n matrix (m >= ). Array Commands find Finds indices of nonzero elements. length Computers number of elements. linspace Creates regularly spaced vector. logspace Creates logarithmically spaced vector. max Returns largest element. min Returns smallest element. prod Product of each column. size Computes array size. sort Sorts each column. sum Sums each column.

4 Example 1 >> X = [ ]; indices = find(x) indices = >> indices = find(x>) indices = 8 9 >> length(x) 9 >> max(x) 8 >> min(x) - > sort(x) >> sum(x) 16 >> sum(x,) 16 >> sum(x,1) >> Example Create a vector of 100 linearly spaced numbers from 1 to 500: A = linspace(1,5); Create a vector of 4 linearly spaced numbers from 1 to 1: >> A = linspace(1,1,4) A = Matrix Functions Much of MATLAB s power comes from its matrix functions. Some useful ones are: eig inv poly det size eigenvalues and eigenvectors inverse characteristic polynomial determinant size

5 [V,D] = eig(a) produces matrices of eigenvalues (D) and eigenvectors (V) of matrix A M = magic(n) returns an n-by-n matrix constructed from the integers 1 through n^ with equal row and column sums. The order n must be a scalar greater than or equal to. >> A=magic() A = >> [c d]=eig(a) c = d = Calculus The Symbolic Math Toolbox provides functions to do the basic operations of calculus; differentiation, limits, integration, summation, and Taylor series expansion. The following sections outline these functions. Differentiation diff(f) differentiates f with respect to its symbolic variable (in this case x) Let s create a symbolic expression. >>syms a x f = sin(a*x) diff(f) f = sin(a*x) a*cos(a*x) To differentiate with respect to the variable a, type diff(f,a) which returns df / da cos(a*x)*x To calculate the second derivatives with respect to x and a, respectively, type diff(f,) or diff(f,x,) %% prefer this formula which return -sin(a*x)*a^

6 Limits The fundamental idea in calculus is to make calculations on functions as a Variable gets close to or approaches a certain value. Recall that the definition of the derivative is given by a limit provided this limit exists. The Symbolic Math Toolbox allows you to compute the limits of functions in a direct manner. >>syms h n x limit( (cos(x+h) - cos(x))/h,h,0 ) -sin(x) And limit( (1 + x/n)^n,n,inf ) exp(x) In the case of undefined limits, the Symbolic Math Toolbox returns NaN (not a number). The command limit(1/x,x,0) or limit(1/x)%% prefer only if function of x only. returns NaN Observe that the default case,limit(f) is the same as limit(f,x,0).explore the options for the limit command in this table. Here, we assume that f is a function of the symbolic object x.

7 Integration If f is a symbolic expression, then the integration of f int(f) xx dddd We can do this in (at least) three different ways. The shortest is: >>int( xˆ ) 1/*xˆ Alternatively, we can define x symbolically first, and then leave off the single quotes in theint statement. >>syms x >>int(xˆ) 1/*xˆ Mathematical Operation MATLAB Command int(x^n) or int(x^n,x) >>int(sin(*x),x,0,pi/) 1 >> g = 'cos(a*t + b)' g = cos(a*t + b) >>int(g) sin(b + a*t)/a 5.Solving Equations Solving Algebraic Equations If S is a symbolic expression,solve(s)attempts to find values of the symbolic variable in S (as determined by findsym) for which S is zero. For example, >>syms a b c x S = a*x^ + b*x + c; solve(s) -(b + (b^ - 4*a*c)^(1/))/(*a) -(b - (b^ - 4*a*c)^(1/))/(*a)

8 This is a symbolic vector whose elements are the two solutions. If you want to solve for a specific variable, you must specify that variable as an additional argument. For example, if you want to solve S for b, use the command >> b = solve(s,b) b = -(a*x^ + c)/x Note that these examples assume equations of the form f(x) = 0. If you need to solve equations of the form f(x)=q(x) you must use quoted strings. In particular, the command s = solve('cos(*x)+sin(x)=1') s = 0 pi/6 (5*pi)/6 Several Algebraic Equations Now let s look at systems of equations. Suppose we have the system xx yy = 0 xx yy = αα and we want to solve for x and y. First create the necessary symbolic objects. syms x y alpha There are several ways to address the output of solve. One is to use a two-output call >>syms x y alpha >> [x,y] = solve(x^*y^, x-y/-alpha) x = alpha 0 y = 0 (-)*alpha Single Differential Equation The function dsolve computes symbolic solutions to ordinary differential equations. The equations are specified by symbolic expressions containing the letter D to denote differentiation. The symbols D, D,... DN, correspond to the second, third,..., Nth derivative, respectively. Thus, Dy is the Symbolic Mathof dddd dddd The dependent variables are those preceded byd and the default independent variable is t. Note that names of symbolic variables should not contain D.The independent variable can be changed from t to some other symbolic variable by including that variable as the last input argument. Initial conditions can be specified by additional equations. If initial conditions are not specified, the solutions contain constants of integration, C1, C, etc. The output from dsolve parallels the output from solve. That is, you can call

9 D solve with the number of out put variables equal to the number of dependent variables or place the output in a structure whose fields contain the solutions of the differential equations. Example 1 The following call to dsolve dsolve('dy=1+y^') uses y as the dependent variable and t as the default independent variable. The output of this command is >>dsolve('dy=1+y^') i -i tan(c4 + t) To specify an initial condition, use y = dsolve('dy=1+y^','y(0)=1') y = tan(t+1/4*pi) Notice that y is in the MATLAB workspace, but the independent variable t is not. Thus, the command diff(y,t) returns an error. To place t in the workspace, type syms t. 6.Linear algebra in MATLAB Solving Equation One of the most important problems in technical computing is the solution of simultaneous linear equations. In matrix notation, this problem can be stated as follows. we may need to find x 1, x, and x so that X 1 + X X = 1 X 1 6X + 4X = X X + X = 1 1 The problem can be rewritten in matrix-vector notation. We introduce a matrix A and a vector b by 1 A = ; 1 b = 1

10 Now we want to find the solution vector [ X X ] A = [ 1-1; ; -1 - ] b = [ 1; -; 1 ] X = A\b Matlab should give the solution X = so that AX = b 1 X X X X 1 = 1 Polynomial Roots and Characteristic Polynomial If p is a row vector containing the coefficients of a polynomial, roots(p) returns a column vector whose elements are the roots of the polynomial. If r is a column vector containing the roots of a polynomial, poly(r) returns a row vector whose elements are the coefficients of the polynomial. To find the roots of following polynomial S S + 1.5S S S S + 15 The polynomial coefficients are entered in a row vector in descending powers. The roots are found using roots. p = [ ] r = roots(p) The polynomial roots are obtained in column vector r = i i i i If we want to find the coefficient of polynomial that has the roots-1, -, - ± j4. We write this r = [ *i --4*i ] p = poly(r) The coefficients of the polynomial equation are obtained in a row vector. p = Therefore, the polynomial equation is

11 Polynomial Evaluation S 4 + 9S + 45S + 87S + 50 = 0 If c is a vector whose elements are the coefficients of a polynomial in descending powers, the polyval(c, x) is the value of the polynomial evaluated at x. For example, to evaluate the above polynomial at points 0, 1,,, and 4, use the commands >> c = [1 1]; x = 0:1:4; y = polyval(c, x) y = You may use the following functions; try to find out its function Polyval,polyvalm (Read the last example in this file). Laplace Transformation The laplace Transformation in MATLAB is very easy way. MATLAB has a function called (laplace) which transfer a function from time-domain to S- Domain. Before you use this function you must declare the variable by syms function,see the below example >>syms t >>laplace(t^5) 10/s^6 To get the laplace inverse it is very easy also, only use ilaplace after declere the variable >>syms s ilaplace(1/(s-1)) exp(t) Or by using Partial-Faction Expansion with MatLab as below example Consider the following transfer function: S + 5S + S + 6 S + 6S + 11S + 6 we write num=[ 5 6] den=[ ] [r,p,k]=residue(num,den) r -6-4

12 p k Which mean 6 + S S + + S + 1 Example: Notes: POLYVAL : Y = polyval(p,x) returns the value of a polynomial P evaluated at X (X is vector or matrix). POLYVALM : Evaluate polynomial with matrix argument (X is vector or matrix).