MATLAB. functions. Overview

Transcription

1 MATLAB Functions 3 Overview In this chapter we start studying the many, many, mathematical functions that are included in MATLAB. Some time will be spent introducing complex variables and how MATLAB handles them. Later we will learn how to write custom (user written) functions which are a special form of m-file. Two dimensional plotting functions will be introduced. We will also explore programming constructs for flow control (if-else-elseif code blocks) and looping (for loop etc.). Next data analysis functions will be investigated, which includes sample statistics and histogram plotting functions. Chapter 3: Overview 3 1

2 Mathematical Functions Common Math Functions Table 3.1: Common math functions Function Description Function Description abs(x) sqrt(x) round(x) nearest integer fix(x) nearest integer floor(x) x nearest integer ceil(x) nearest integer toward toward sign(x) rem(x,y) the remainder 1, x < 0 of x y 0, x = 0 1, x > 0 x exp(x) log(x) natural log e x lnx log10(x) log base 10 log 10 x Examples:» x = [ ];» round(x) ans = -6 6» fix(x) ans = -5 5» floor(x) ans = -6 5» ceil(x) Chapter 3: Mathematical Functions 3

3 ans = -5 6» sign(x) ans = -1 1» rem(3,6) ans = 5 Trigonometric and Hyperbolic Functions Unlike pocket calculators, the trigonometric functions always assume the input argument is in radians The inverse trigonometric functions produce outputs that are in radians Table 3.: Trigonometric functions Function Description Function Description sin(x) tan(x) acos(x) atan(y, x) sin( x) tan( x) cos(x) asin(x) atan(x) cos 1 ( x) tan 1 ( x) the inverse tangent of y x including the correct quadrant cos( x) sin 1 ( x) Examples: A simple verification that sin ( x) + cos ( x) = 1 Chapter 3: Mathematical Functions 3 3

4 » x = 0:pi/10:pi;» [x' sin(x)' cos(x)' (sin(x).^+cos(x).^)'] ans = Parametric plotting: Verify that by plotting sinθ versus cosθ for 0 θ πwe obtain a circle» theta = 0:*pi/100:*pi; % Span 0 to pi with 100 pts» plot(cos(theta),sin(theta)); axis('square')» title('parametric Plotting','fontsize',18)» ylabel('y - axis','fontsize',14)» xlabel('x - axis','fontsize',14)» figure()» hold Current plot held» plot(cos(5*theta).*cos(theta),... cos(5*theta).*sin(theta)); % A five-leaved rose Chapter 3: Mathematical Functions 3 4

5 1 0.8 Parametric Plotting Circle Rose 0. y - axis x - axis The trig functions are what you would expect, except the features of atan(y,x) may be unfamiliar y y 4 x x φ 1 = φ =.0344 Chapter 3: Mathematical Functions 3 5

6 » [atan(/4) atan(,4)] ans = % the same» [atan(-4/-) atan(-4,-)] ans = % different; why? e x The hyperbolic functions are defined in terms of There are no special concerns in the use of these functions except that atanh requires an argument that must not exceed an absolute value of one Complex Number Functions Table 3.3: Hyperbolic functions Function Description Function Description sinh(x) tanh(x) acosh(x) e x e x cosh(x) asinh(x) e x e x ln( x + x + 1 ) e x + e x ln( x+ x 1) atanh(x) Before discussing these functions we first review a few facts about complex variable theory In electrical engineering complex numbers appear frequently A complex number is an ordered pair of real numbers 1 e x e x x ln , x 1 1 x 1.Tom M. Apostle, Mathematical Analysis, second edition, Addison Wesley, p. 15, Chapter 3: Mathematical Functions 3 6

7 denoted The first number, a, is called the real part, while the second number, b, is called the imaginary part For algebraic manipulation purposes we write ( ab, ) = a + ib = a+ jb where i = j = 1 ; electrical engineers typically use j since i is often used to denote current Note: ( ab, ) 1 1 = 1 j j = 1 For complex numbers we define/calculate = a 1 + jb 1 and z = a + jb + z = ( a 1 + a ) + jb ( 1 + b ) (sum) z = ( a 1 a ) + jb ( 1 b ) (difference) z = ( a 1 a b 1 b ) + ja ( 1 b + b 1 a ) (product) ---- z = ( a 1 a + b 1 b ) ja ( 1 b b 1 a ) a + b (quotient) = a 1 + b 1 (magnitude) = tan 1 ( b 1 a 1 ) (angle) = a 1 jb 1 (complex conjugate) MATLAB is consistent with all of the above, starting with the fact that i and j are predefined to be 1 Chapter 3: Mathematical Functions 3 7

8 The rectangular form of a complex number is as defined above, The corresponding polar form is z = ( ab, ) = a+ jb z = r θ where r = a + b and θ = tan 1 ( b a) Imaginary Axis Complex Plane b r θ a Real Axis MATLAB has five basic complex functions, but in reality most all of MATLAB s functions accept complex arguments and deal with them correctly Function conj(x) real(x) imag(x) Table 3.4: Basic complex functions Description Computes the conjugate of which is z = a jb Extracts the real part of which is real( z) = a Extracts the imaginary part of z = a+ jb which is imag( z) z = a+ jb z = a + jb = b Chapter 3: Mathematical Functions 3 8

9 Function Table 3.4: Basic complex functions Description angle(x) computes the angle of z = a+ jb using atan which is atan(imag(z),real(z)) Euler s Formula: A special mathematical result of, special importance to electrical engineers, is the fact that Turning (3.1) around yields It also follows that where e jb = cosb+ jsinb sinθ cosθ = = e jθ e jθ j e jθ e jθ z = a + jb = re jθ (3.1) (3.) (3.3) (3.4) r = a + b, θ = tan 1b ā -, a = rcosθ, b = rsinθ Chapter 3: Mathematical Functions 3 9

10 Some examples:» z1 = +j*4; z = -5+j*7;» [z1 z] ans = i i» [real(z1) imag(z1) abs(z1) angle(z1)] ans = » [conj(z1) conj(z)] ans = i i» [z1+z z1-z z1*z z1/z] ans = i i i i Polar Plots: When dealing with complex numbers we often deal with the polar form. The plot command which directly plots vectors of magnitude and angle is polar(theta,r) This function is also useful for general plotting As an example the equation of a cardioid in polar form, with parameter a, is r = a1 ( + cosθ), 0 θ π» theta = 0:*pi/100:*pi; % Span 0 to pi with 100 pts» r = 1+cos(theta);» polar(theta,r) Chapter 3: Mathematical Functions 3 10

11 Polar Plot Chapter 3: Mathematical Functions 3 11