ECE Lesson Plan - Class #3 Script M-files, Program Control, and Complex Numbers

Transcription

1 ECE Lesson Plan - Class #3 Script M-files, Program Control, and Complex Numbers Script M-files : A script file is an external file that contains a sequence of MATLAB. By typing the filename, subsequent MATLAB input is obtained from the file. Script files have a filename extension of.m and are often called M-files. Scripts are the simplest kind of M-file. They are useful for automating blocks of MATLAB commands, such as computations you have to perform repeatedly from the command line. Scripts can operate on existing data in the workspace, or they can create new data on which to operate. Although scripts do not return output arguments, any variables that they create remain in the workspace so you can use them in further computations. In addition, scripts can produce graphical output using commands like plot. Scripts can contain any series of MATLAB and require no declarations. Like any M-file, scripts can contain comments. Any text following a percent sign (%) on a given line is comment text. Comments can appear on lines by themselves, or you can app them to the of any executable line. Enter the word 'edit' at the MATLAB prompt to open the MATLAB Editor. fprintf Write formatted data to file or screen fprintf([fid],format,a,...) fprintf([fid], format,a,...) writes to file pointed to by FID which is obtained by fopen. If FID is omitted output is sent to standard output--the screen. The format string specifies notation, alignment, significant digits, field width, and other aspects of output format. It can contain ordinary alphanumeric characters; along with escape characters, conversion specifiers, and other characters, organized as shown below: Tables The following tables describe the non-alphanumeric characters found in format specification strings. Character \n New line 1

2 \t Horizontal tab \b Backspace \r Carriage return \f Form feed \\ Backslash \'' or '' (two single quotes) Single quotation mark %% Percent character Conversion characters specify the notation of the output. These characters are also known as specifiers. Specifier %c Single character %d Decimal notation (signed) %e Exponential notation (using a lowercase e as in e+00) %E Exponential notation (using an uppercase E as in E+00) %f Fixed-point notation %g The more compact of %e or %f, as defined in [2]. Insignificant zeros do not print. %G Same as %g, but using an uppercase E %o Octal notation (unsigned) %s String of characters %u Decimal notation (unsigned) %x Hexadecimal notation (using lowercase letters a-f) %X Hexadecimal notation (using uppercase letters A-F) Other characters can be inserted into the conversion specifier between the % and the conversion character. Character Example A minus sign (-) Left-justifies the converted argument in its field. %-5.2d A plus sign (+) Always prints a sign character (+ or -). %+5.2d Zero (0) Pad with zeros rather than spaces. %05.2d Digits (field width) A digit string specifying the minimum number of digits to be printed. %6f Digits (precision) A digit string including a period (.) specifying the number of digits to be printed to the right of the decimal point. %6.2f Examples The command >> fprintf('a unit circle has circumference %g.\n',2*pi) displays a line on the screen: A unit circle has circumference

3 To insert a single quotation mark in a string, use two single quotation marks together. For example, >> fprintf('it''s Friday.\n') displays on the screen: It's Friday. The commands >> B = [ ; ] >> fprintf('x is %6.2f meters or %8.3f mm\n',9.9,9900,b) display the lines: X is 9.90 meters or mm X is 8.80 meters or mm X is 7.70 meters or mm Explicitly convert MATLAB double-precision variables to integral values for use with an integral conversion specifier. For instance, to convert decimal to hexadecimal format: >> a = [ ]; >> whos % Notice that the class for a is double array. >> fprintf('%x\n',a) 6 A E 2C input Request user input : user_entry = input('prompt') user_entry = input('prompt','s') : The response to the input prompt can be any MATLAB expression, which is evaluated using the variables in the current workspace. user_entry = input('prompt') displays prompt as a prompt on the screen, waits for input from the keyboard, and returns the value entered in user_entry. Remarks: If you press the Return key without entering anything, input returns an empty matrix. I recomm leaving an "arrowhead" or "pointer" at the of your string. An example would be: >> Variable = input('please enter Yadda Yadda Yadda here -> '); 3

4 The white space to the left of the "arrowhed" or "pointer" will indicate to the user where the data needs to be entered. Like this: Example: >>TestInput=('Please enter in a value for the variable TestInput -> ') >> whos % Notice that TestInput is now in the MATLAB workspace. error Display error messages error('error_message') error('error_message') displays an error message and returns control to the keyboard. The error message contains the input string error_message. The error command has no effect if error_message is a null string. Examples The error command provides an error return from M-files. >> XX = 2; >> if XX == 2 >> error('wrong number') >> warning Display warning message warning('message') warning on warning off warning backtrace warning('message') displays the text 'message' as does the disp function, except that with warning, message display can be suppressed. warning off suppresses all subsequent warning messages. warning on re-enables them. warning backtrace is the same as warning on except that the file and line number that produced the warning are displayed. 4

5 pause Halt execution temporarily. pause pause(n) pause on pause off pause, by itself, causes M-files to stop and wait for you to press any key before continuing. pause(n) pauses execution for n seconds before continuing. pause on allows subsequent pause commands to pause execution. pause off ensures that any subsequent pause or pause(n) do not pause execution. >> pause(10) will pause the execution of the MATLAB program for 10 seconds. pause by itself should almost always be preceded with >> disp('press any key to continue.') so that the user knows that the MATLAB Command window is in a pause mode. 5

6 Program Control MATLAB supports standard program control functions including loops and conditionals. These may be used in script M-files or functions. Relational Operators < > <= >= == ~= Relational operations A < B A > B A <= B A >= B A == B A ~= B The relational operators are <,, >,, ==, and ~=. Relational operators perform element-by-element comparisons between two arrays. They return an array of the same size, with elements set to logical true (1) where the relation is true, and elements set to logical false (0) where it is not. The operators <,, >, and use only the real part of their operands for the comparison. The operators == and ~= test real and imaginary parts. The relational operators have precedence midway between the logical operators and the arithmetic operators. To test if two strings are equivalent, use strcmp, which allows vectors of dissimilar length to be compared. Examples If one of the operands is a scalar and the other a matrix, the scalar expands to the size of the matrix. For example, the two pairs of : >> X = 5; >> X >= [1 2 3; 4 5 6; ] >> X = 5*ones(3,3); >> X >= [1 2 3; 4 5 6; ] produce the same result: ans = Logical Operators Logical operations 6

7 A & B A B ~A The symbols &,, and ~ are the logical operators and, or, and not. They work element-wise on arrays, with 0 representing logical false (F), and anything nonzero representing logical true (T). The & operator does a logical and, the operator does a logical or, and ~A complements the elements of A. The function xor(a,b) implements the exclusive or operation. Truth tables for these operators and functions follow. Inputs and or xor not A B A&B A B xor(a,b) ~A The logical operators have the lowest precedence, with arithmetic operators and relational operators being evaluated first. The precedence for the logical operators with respect to each other is: 1. not has the highest precedence. 2. and and or have equal precedence, and are evaluated from left to right. Remarks The logical operators have M-file function equivalents, as shown: and A&B and(a,b) or A B or(a,b) not ~A not(a) Examples Here are two scalar expressions that illustrate precedence relationships for arithmetic, relational, and logical operators: >> 1 & >> 3 > 4 & 1 They evaluate to 1 and 0 respectively, and are equivalent to: >> 1 & (0 + 3) >> (3 > 4) & 1 Here are two examples that illustrate the precedence of the logical operators to each other: >> 1 0 & 0 % Should = 0 >> 0 & 0 1 % Should = 1 7

8 for Repeat a specific number of times for variable = expression The general format is for variable = expression statement... statement The columns of the expression are stored one at a time in the variable while the following, up to the, are executed. In practice, the expression is almost always of the form scalar : scalar, in which case its columns are simply scalars. The scope of the for statement is always terminated with a matching. Notice that the matrix 'a' uses the zeros function to preallocate the matrix to conserve memory: Create a Script M-File here called ForExample.m. >> n = 7; >> a = zeros(n) % Preallocate matrix >> for i = 1:n >> for j = 1:n >> a(i, j) = 1/(i+j) >> >> >> a Terminate for, while, switch, and if or indicate last index while expression % (or if, or for) 8

9 B = A(index:,index) is used to terminate for, while, and if. Without an statement, for, while, and if wait for further input. Each is paired with the closest previous unpaired for, while, or if. The command can also serve as the last index in an indexing expression. In that context, = (size(x,k)) when used as part of the kth index. Examples of this use are X(3:) and X(1,1:2:-1). When using to grow an array, as in X(+1)=5. Examples This example shows used with for and if. Indentation provides easier readability. for i = 1:n if a(i) == 0 a(i) = a(i) + 2; Here, is used in an indexing expression: >> A = [ ]; >> B = A(1,7:) In this example, B is a 1-by-3 vector equal to [7 8 9]. if Conditionally execute if expression if expression1 elseif expression2 else if conditionally executes. The simple form is: if expression 9

10 More complicated forms use else or elseif. Each if must be paired with a matching. Arguments A MATLAB expression, usually consisting of smaller expressions or variables joined by relational operators (==, <, >, <=, >=, or ~=). Two examples are: expression count < limit and (height - offset) >= 0. Expressions may also include logical functions, as in: isreal(a). Simple expressions can be combined by logical operators (&,,~) into compound expressions such as: (count < limit) & ((height - offset) >= 0). One or more MATLAB to be executed only if the expression is true (or nonzero). See Examples for information about how nonscalar variables are evaluated. Expressions are evaluated as false unless every element-wise comparison evaluates as true. Thus, given matrices A and B: A = B = The expression: A < B Evaluates as false Since A(1,1) is not less than B(1,1). A < (B+1) Evaluates as true Since no element of A is greater than the corresponding element of B. A & B Evaluates as false Since A(1,2) B(1,2) is false. 5 > B Evaluates as true Since every element of B is less than 5. else Conditionally execute if expression else The else command is used to delineate an alternate block of. if expression else 10

11 The second set of is executed if the expression has any zero elements. The expression is usually the result of expression rop expression where rop is ==, <, >, <=, >=, or ~=. elseif Conditionally execute if expression elseif expression The elseif command conditionally executes. if expression elseif expression The second block of executes if the first expression has any zero elements and the second expression has all nonzero elements. The expression is usually the result of expression rop expression where rop is ==, <, >, <=, >=, or ~=. else if, with a space between the else and the if, differs from elseif, with no space. The former introduces a new, nested, if, which must have a matching. The latter is used in a linear sequence of conditional with only one terminating. The two segments if A if A x = a x = a else elseif B if B x = b x = b elseif C else x = c if C else x = c x = d else x = d 11

12 produce identical results. Exactly one of the four assignments to x is executed, deping upon the values of the three logical expressions, A, B, and C. Examples Here is an example showing if, else, and elseif: Create a Script M-File here called ElseIfExample.m: >> for i = 1:8 >> for j = 1:8 >> if i == j >> a(i,j) = 2; >> elseif abs([i j]) == 1 >> a(i,j) = 1; >> else >> a(i,j) = 0; >> >> >> while Repeat an indefinite number of times while expression while repeats an indefinite number of times. The are executed while the real part of expression has all nonzero elements. expression is usually of the form expression rop expression where rop is ==, <, >, <=, >=, or ~=. The scope of a while statement is always terminated with a matching. break Terminate execution of for or while loop break 12

13 break terminates the execution of for and while loops. In nested loops, break exits from the innermost loop only. Examples The indented are repeatedly executed until nonpositive n is entered. Create a Script M-File here called BreakExample.m >> while 1 >> n = input('enter n. n <= 0 quits. n = ') >> if n <= 0,break, >> r = (magic(n)) >> >> disp('breakexample.m has stopped.') % How many iterations required for a Binary Search on a given number? Create a Script M-File here called BinarySearch.m >> BinSrch = input('please input any number -> '); >> COUNTER = 1; >> for i = 1:1000 % Assume "Binary Search" is < >> BinSrch = BinSrch/2; >> if BinSrch <= 1 >> break >> >> COUNTER = COUNTER + 1; >> >> COUNTER 13

14 Functions for Complex Numbers i,j Imaginary Unit i x+yi x+i*y As the basic imaginary unit sqrt(-1), i is used to enter complex numbers. Since i is a built-in value of sqrt(-1), it can be overridden and used as a variable. This permits you to use i as an index in for loops, etc. If desired, use the character i without a multiplication sign as a suffix in forming a complex numerical constant. You can also use the character j as the imaginary unit as if it were i. >> i >> j Let's prove that, for example, the expressions 3+2i, 3+2*i, 3+2i, 3+2*j and 3+2*sqrt(-1) all have the same value. >> 3+2i >> 3+2*i >> 3+2j >> 3+2*j >> 3+2*sqrt(-1) % Changing from rectangular coordinates to polar coordinates. >> X = 3; >> Y = 4; >> R = abs(x+i*y); >> THETA = angle(x+i*y); >> R*exp(i*THETA) % Notice that the output here is rectangular. For complex Z = X + i*y, exp returns the complex exponential. Or said in a question: Does EXP(Z) = EXP(X)*(COS(Y)+i*SIN(Y))? >> X = 5; >> Y = 5; >> Z = X + Y*i >> exp(z) - (exp(x)*(cos(y)+i*sin(y))) 14

15 % If this equals 0 then the two sides of the equation are equal. real Real part of complex number X = real(z) X = real(z) returns the real part of the elements of the complex number Z. >> real(2+3*i) >> Z >> real(z) >> R*exp(i*THETA) >> real(r*exp(i*theta)) imag Imaginary part of a complex number Y = imag(z) Y = imag(z) returns the imaginary part of the elements of complex number Z. >> imag(2+3i) >> Z >> imag(z) >> R*exp(i*THETA) >> imag(r*exp(i*theta)) abs Complex Magnitude Y = abs(x) 15

16 abs(x) returns the complex modulus (magnitude): abs(x) = sqrt(real(x).^2 + imag(x).^2) Examples >> abs(3+4i) >> abs(2+2i) >> abs(1+i) Notice that the multiplication sign(*) is not necessary for "i" if the factor is first. angle Phase angle P = angle(z) P = angle(z)returns the phase angle(s), in radians, of complex number Z. To convert complex Z (in rectangular form -> x + iy) to polar form, solve for the magnitude and phase by using: >> R = abs(z) % The magnitude of complex number Z. (FYI - this value is unitless) >> theta = angle(z) % The phase angle, in radians, of complex number The conversion that is taking place is as follows: Z = X +(Y*i) = R.*exp(i*theta) Algorithm angle is calculated using: angle(z) = atan2(imag(z),real(z)) >> ZZ = [1+i 4-9i 3+8i 2-5i; 2+2i 6-8i 5+4i 4-i; 8+7i 6-6i 7+7i 5-5i; 9+i 8-2i 7+3i 6-4i] >> Theta = angle(zz) >> D = Theta * % 1 Radian = Degrees conj Complex conjugate ZC = conj(z) 16

17 ZC = conj(z) returns the complex conjugate of the elements of Z. Algorithm If Z is a complex number of the form X + iy or R*exp(iTHETA): conj(z) = real(z) - i*imag(z) >> ZZ >> conj(zz) >> RR = 5; >> THETA1 = ; >> CP = RR*exp(i*THETA1) % Notice that Cartesian (rectangular) coordinates are returned. >> conj(cp) >> CPneg = RR*exp(-i*THETA1) >> conj(cpneg) ctranspose ' Complex conjugate transpose. X' is the complex conjugate transpose of X. B = CTRANSPOSE(A) is called for the syntax A' that results in the complex conjugate transpose of A. >> CP' >> ctranspose(cp) >> W = [7+9i 5-6i 3+4i 7-6i] >> W' % Conjugate and Transpose >> W.' % Only Transposes >> ctranspose(w) % Equals W' cart2pol CART2POL Transform Cartesian to polar coordinates. [THETA,R] = CART2POL(X,Y) transforms corresponding elements of data stored in Cartesian coordinates X,Y to polar coordinates (angle θ and radius R). The arrays X and Y must be the same size (or either can be scalar). θ is returned in radians. [THETA,R] = cart2pol(x,y) 17

18 [THETA,R] = cart2pol(x,y)transforms two-dimensional Cartesian coordinates stored in corresponding elements of arrays X and Y into polar coordinates. Examples >> X = 3; % The Real Part >> Y = 4; % The Imaginary Part >> [THETA, R] = cart2pol(x,y) >> X = 1; >> Y = 1; >> [THETA, R] = cart2pol(x,y) >> THETA * pol2cart Transform polar coordinates to Cartesian coordinates. [X,Y] = pol2cart(theta,r) [X,Y] = pol2cart(theta,r) transforms the polar coordinate data stored in corresponding elements of THETA and R (magnitude) to 2-dimensional Cartesian, or x-y, coordinates. The arrays THETA and R must be the same size (or either can be scalar). The values in THETA must be in radians. >> [X,Y] = pol2cart( ,5) >> [X,Y] = pol2cart(pi/4,5.6569) % Factors of pi work best for phase inputs. polar Plot polar coordinates polar(theta,r) polar(theta,r,linespec) The polar function accepts polar coordinates, plots them in a Cartesian plane, and draws the polar grid on the plane. polar(theta,r) creates a polar coordinate plot of the angle theta versus the radius r. theta is the angle from the x-axis to the radius vector specified in radians; r is the length of the radius vector specified in dataspace units. 18

19 polar(theta,rho,linespec) LineSpec specifies the line type, plot symbol, and color for the lines drawn in the polar plot. Create a simple polar plot using a dashed, red line: >> t1 = 0:.01:2*pi; >> polar(t1,sin(2*t1).*cos(2*t1),'--r') >> hold on >> polar(t1+0.25,sin(2*t1).*cos(2*t1),'*b') >> polar(t1+0.5,sin(2*t1).*cos(2*t1),'.-g') References: - The MathWorks, Inc. Online (Helpdesk) Documentation by The MathWorks, Inc. - Mastering MATLAB 5 by Duane Hanselman and Bruce Littlefield - Engineering Problem Solving with MATLAB by D.M. Etter - Using MATLAB Version 5.0 by The MathWorks, Inc. - Getting Started with MATLAB Version 5.0 by The MathWorks, Inc. 19