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This document provides an overview of mathematical functions in MATLAB, including: 1) Common math functions such as absolute value, rounding, floor/ceiling, exponents, logs, and trigonometric functions. 2) How to write custom functions and use programming constructs like if/else statements and for loops. 3) Data analysis functions including statistics and histograms. 4) Complex number representation and basic complex functions in MATLAB.
![ECE 1010 ECE Problem Solving I
Mathematical Functions
Common Math Functions
Table 3.1: Common math functions
Function Description Function Description
abs(x) x sqrt(x) x
round(x) nearest integer fix(x) nearest integer
floor(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
exp(x) x log(x) natural log ln x
e
log10(x) log base 10
log 10 x
Examples:
» x = [-5.5 5.5];
» round(x)
ans = -6 6
» fix(x)
ans = -5 5
» floor(x)
ans = -6 5
» ceil(x)
Chapter 3: Mathematical Functions 3–2](https://image.slidesharecdn.com/1010n3a-120828092928-phpapp02/85/1010n3a-2-320.jpg)
![ECE 1010 ECE Problem Solving I
» x = 0:pi/10:pi;
» [x' sin(x)' cos(x)' (sin(x).^2+cos(x).^2)']
ans =
0 0 1.0000 1.0000
0.3142 0.3090 0.9511 1.0000
0.6283 0.5878 0.8090 1.0000
0.9425 0.8090 0.5878 1.0000
1.2566 0.9511 0.3090 1.0000
1.5708 1.0000 0.0000 1.0000
1.8850 0.9511 -0.3090 1.0000
2.1991 0.8090 -0.5878 1.0000
2.5133 0.5878 -0.8090 1.0000
2.8274 0.3090 -0.9511 1.0000
3.1416 0.0000 -1.0000 1.0000
• Parametric plotting:
– Verify that by plotting sin θ versus cos θ for 0 ≤ θ ≤ 2π we
obtain a circle
» theta = 0:2*pi/100:2*pi; % Span 0 to 2pi 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(2)
» 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](https://image.slidesharecdn.com/1010n3a-120828092928-phpapp02/85/1010n3a-4-320.jpg)
![ECE 1010 ECE Problem Solving I
» [atan(2/4) atan2(2,4)]
ans = 0.4636 0.4636 % the same
» [atan(-4/-2) atan2(-4,-2)]
ans = 1.1071 -2.0344 % different; why?
x
• The hyperbolic functions are defined in terms of e
Table 3.3: Hyperbolic functions
Function Description Function Description
sinh(x) x –x cosh(x) x –x
e –e e –e
-----------------
- -----------------
-
2 2
tanh(x) x –x asinh(x) 2
e –e
------------------ ln ( x + x + 1 )
x –x
e +e
acosh(x) 2 atanh(x) 1+x
ln ( x + x – 1 ) ln -----------, x ≤ 1
-
1–x
• 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
• 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 numbers1
1. Tom M. Apostle, Mathematical Analysis, second edition, Addison Wesley,
p. 15, 1974.
Chapter 3: Mathematical Functions 3–6](https://image.slidesharecdn.com/1010n3a-120828092928-phpapp02/85/1010n3a-6-320.jpg)
![ECE 1010 ECE Problem Solving I
• Some examples:
» z1 = 2+j*4; z2 = -5+j*7;
» [z1 z2]
ans =
2.0000 + 4.0000i -5.0000 + 7.0000i
» [real(z1) imag(z1) abs(z1) angle(z1)]
ans =
2.0000 4.0000 4.4721 1.1071
» [conj(z1) conj(z2)]
ans =
2.0000 - 4.0000i -5.0000 - 7.0000i
» [z1+z2 z1-z2 z1*z2 z1/z2]
ans =
-3.0000 +11.0000i 7.0000 - 3.0000i
-38.0000 - 6.0000i 0.2432 - 0.4595i
Polar Plots: When dealing with complex numbers we often deal
with the polar form. The plot command which directly plots vec-
tors 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 = a ( 1 + cos θ ), 0 ≤ θ ≤ 2π
» theta = 0:2*pi/100:2*pi; % Span 0 to 2pi with 100 pts
» r = 1+cos(theta);
» polar(theta,r)
Chapter 3: Mathematical Functions 3–10](https://image.slidesharecdn.com/1010n3a-120828092928-phpapp02/85/1010n3a-10-320.jpg)
1010n3a
- 1. ECE 1010 ECE Problem Solving I MATLAB 3 Functions 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 plot- ting functions. Chapter 3: Overview 3–1
- 2. ECE 1010 ECE Problem Solving I Mathematical Functions Common Math Functions Table 3.1: Common math functions Function Description Function Description abs(x) x sqrt(x) x round(x) nearest integer fix(x) nearest integer floor(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 exp(x) x log(x) natural log ln x e log10(x) log base 10 log 10 x Examples: » x = [-5.5 5.5]; » round(x) ans = -6 6 » fix(x) ans = -5 5 » floor(x) ans = -6 5 » ceil(x) Chapter 3: Mathematical Functions 3–2
- 3. ECE 1010 ECE Problem Solving I ans = -5 6 » sign(x) ans = -1 1 » rem(23,6) 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 3.2: Trigonometric functions Function Description Function Description sin(x) sin ( x ) cos(x) cos ( x ) tan(x) tan ( x ) asin(x) –1 sin ( x ) acos(x) –1 atan(x) –1 cos ( x ) tan ( x ) atan2(y, the inverse tan- x) gent of y ⁄ x including the correct quad- rant Examples: 2 2 • A simple verification that sin ( x ) + cos ( x ) = 1 Chapter 3: Mathematical Functions 3–3
- 4. ECE 1010 ECE Problem Solving I » x = 0:pi/10:pi; » [x' sin(x)' cos(x)' (sin(x).^2+cos(x).^2)'] ans = 0 0 1.0000 1.0000 0.3142 0.3090 0.9511 1.0000 0.6283 0.5878 0.8090 1.0000 0.9425 0.8090 0.5878 1.0000 1.2566 0.9511 0.3090 1.0000 1.5708 1.0000 0.0000 1.0000 1.8850 0.9511 -0.3090 1.0000 2.1991 0.8090 -0.5878 1.0000 2.5133 0.5878 -0.8090 1.0000 2.8274 0.3090 -0.9511 1.0000 3.1416 0.0000 -1.0000 1.0000 • Parametric plotting: – Verify that by plotting sin θ versus cos θ for 0 ≤ θ ≤ 2π we obtain a circle » theta = 0:2*pi/100:2*pi; % Span 0 to 2pi 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(2) » 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. ECE 1010 ECE Problem Solving I Parametric Plotting 1 Circle 0.8 0.6 0.4 Rose 0.2 y - axis 0 -0.2 -0.4 -0.6 -0.8 -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x - axis • The trig functions are what you would expect, except the fea- tures of atan2(y,x) may be unfamiliar y y 4 2 x 2 x φ 1 = 0.4636 4 φ 2 = – 2.0344 Chapter 3: Mathematical Functions 3–5
- 6. ECE 1010 ECE Problem Solving I » [atan(2/4) atan2(2,4)] ans = 0.4636 0.4636 % the same » [atan(-4/-2) atan2(-4,-2)] ans = 1.1071 -2.0344 % different; why? x • The hyperbolic functions are defined in terms of e Table 3.3: Hyperbolic functions Function Description Function Description sinh(x) x –x cosh(x) x –x e –e e –e ----------------- - ----------------- - 2 2 tanh(x) x –x asinh(x) 2 e –e ------------------ ln ( x + x + 1 ) x –x e +e acosh(x) 2 atanh(x) 1+x ln ( x + x – 1 ) ln -----------, x ≤ 1 - 1–x • 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 • 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 numbers1 1. Tom M. Apostle, Mathematical Analysis, second edition, Addison Wesley, p. 15, 1974. Chapter 3: Mathematical Functions 3–6
- 7. ECE 1010 ECE Problem Solving I denoted ( a, b ) – 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 ( a, b ) = a + ib = a + jb where i = j = – 1 ; electrical engineers typically use j since i is often used to denote cur- rent Note: –1 × –1 = –1 ⇒ j × j = –1 • For complex numbers z 1 = a 1 + jb 1 and z 2 = a 2 + jb 2 we define/calculate z1 + z2 = ( a1 + a2 ) + j ( b1 + b2 ) (sum) z 1 – z 2 = ( a 1 – a 2 ) + j ( b 1 – b 2 ) (difference) z 1 z 2 = ( a 1 a 2 – b 1 b 2 ) + j ( a 1 b 2 + b 1 a 2 ) (product) z1 ( a1 a2 + b1 b2 ) – j ( a1 b2 – b1 a2 ) ---- = --------------------------------------------------------------------------- (quotient) - z2 2 2 a +b 2 2 2 2 z1 = a 1 + b 1 (magnitude) –1 ∠z 1 = tan ( b 1 ⁄ a 1 ) (angle) z 1∗ = 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. ECE 1010 ECE Problem Solving I • The rectangular form of a complex number is as defined above, z = ( a, b ) = a + jb • The corresponding polar form is z = r ∠θ 2 2 –1 where r = a + b and θ = tan ( b ⁄ a ) Imaginary Axis Complex Plane b r θ Real a Axis • MATLAB has five basic complex functions, but in reality most all of MATLAB’s functions accept complex arguments and deal with them correctly Table 3.4: Basic complex functions Function Description conj(x) Computes the conjugate of z = a + jb which is z∗ = a – jb real(x) Extracts the real part of z = a + jb which is real ( z ) = a imag(x) Extracts the imaginary part of z = a + jb which is imag ( z ) = b Chapter 3: Mathematical Functions 3–8
- 9. ECE 1010 ECE Problem Solving I Table 3.4: Basic complex functions Function Description angle(x) computes the angle of z = a + jb using atan2 which is atan2(imag(z),real(z)) Euler’s Formula: A special mathematical result of, special importance to electrical engineers, is the fact that jb e = cos b + j sin b (3.1) • Turning (3.1) around yields jθ – jθ e –e sin θ = --------------------- - (3.2) 2j jθ – jθ e +e cos θ = --------------------- - (3.3) 2 • It also follows that jθ z = a + jb = re (3.4) where 2 2 –1 b r= a + b , θ = tan --, a = r cos θ, b = r sin θ - a Chapter 3: Mathematical Functions 3–9
- 10. ECE 1010 ECE Problem Solving I • Some examples: » z1 = 2+j*4; z2 = -5+j*7; » [z1 z2] ans = 2.0000 + 4.0000i -5.0000 + 7.0000i » [real(z1) imag(z1) abs(z1) angle(z1)] ans = 2.0000 4.0000 4.4721 1.1071 » [conj(z1) conj(z2)] ans = 2.0000 - 4.0000i -5.0000 - 7.0000i » [z1+z2 z1-z2 z1*z2 z1/z2] ans = -3.0000 +11.0000i 7.0000 - 3.0000i -38.0000 - 6.0000i 0.2432 - 0.4595i Polar Plots: When dealing with complex numbers we often deal with the polar form. The plot command which directly plots vec- tors 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 = a ( 1 + cos θ ), 0 ≤ θ ≤ 2π » theta = 0:2*pi/100:2*pi; % Span 0 to 2pi with 100 pts » r = 1+cos(theta); » polar(theta,r) Chapter 3: Mathematical Functions 3–10
- 11. ECE 1010 ECE Problem Solving I 90 Polar Plot 2 120 60 1.5 150 1 30 0.5 180 0 210 330 240 300 270 Chapter 3: Mathematical Functions 3–11