Octave Fundamentals
SIGSLAB I
Octave Fundamentals
I. Objectives:
-
To be able to familiarize with some of the commands and capabilities of Octave.
-
To be able to learn how to do Matrix/Vector manipulation, complex algebra, and graphics in Octave.
II. Introduction
GNU Octave is a high-level interpreted language, primarily intended for numerical computations. It provides capabilities for the numerical solution of linear and nonlinear problems, and for performing other numerical experiments. It also provides extensive graphics capabilities for data visualization and manipulation. Working with the Octave environment is straightforward because most commands are entered as you write them mathematically.
from IPython.display import IFrame
IFrame('https://www.gnu.org/software/octave/', width='100%', height=540)
<iframe
width="100%"
height="540"
src="https://www.gnu.org/software/octave/"
frameborder="0"
allowfullscreen
></iframe>
2.1 Loading Octave in iPython Notebook
%load_ext oct2py.ipython
2.2 Running Octave in a cell
%%octave
x=exp(-0.2696*.2)*sin(2*pi*0.2)/(0.01*sqrt(3)*log(18))
x = 18.000
The result is displayed above as x = 18.000.
If you want to know the value of an expression without the variable, you can simply type the expression by itself, e.x.
%%octave
4/3
ans = 1.3333
If you wish to create a new variable but do not want to see the Octave response, type a semicolon at the end of the expression, e.x.
%%octave
a = 7 + 5;
The semicolon is very useful when large vectors or matrices are defined or computed in intermediate step of a computation. You can check the value of variable at any time by entering the variable name.
%%octave
a = 7 + 5;
a
a = 12
2.3 Output formatting
%%octave
printf("Area = %7.3f Square meters \n",pi*4.5^2)
Area = 63.617 Square meters
The %7.3f prints a floating point number at least 7 characters wide, with 3 digits after the decimal point . The sequence \n advances the output to the left margin on the next line
%%octave
a = pi
disp(sprintf('2 decimals: %0.2f', a))
disp(sprintf('6 decimals: %0.6f', a))
format long
a
format short
a
a = 3.1416
2 decimals: 3.14
6 decimals: 3.141593
a = 3.14159265358979
a = 3.1416
2.4 Character String
%%octave
c = "Good"
c = Good
A text variable can be augmented with more text variable, e.x.
%%octave
c = "Good"
cs = [ c, "luck"]
c = Good
cs = Goodluck
2.5 Elementary Matrix Operation
%%octave
A = [1 2; 3 4]
A =
1 2
3 4
The entire row or column of a matrix can be addressed by means of the symbol (:), e.x.
%%octave
A = [1 2; 3 4];
row2 = A(2,:)
row2 =
3 4
Similarly, the statement A( : , 2) addresses all elements of the second column in A.
%%octave
A = [1 2; 3 4];
col2 = A(:,2)
col2 =
2
4
Typing A(:) will return all the elements of A in one column.
%%octave
A = [1 2; 3 4];
B = A(:)
size(B)
B =
1
3
2
4
ans =
4 1
Matrices of the same dimension can be added or subtracted.
%%octave
A = [1 2; 3 4];
B = [4 3; 2 1];
C = A + B
C =
5 5
5 5
Two matrices, A and B, can be multiplied together to form the product AB if they are comformable.
%%octave
A = [1 2; 3 4];
B = [1 2 3; 4 5 6];
A * B
ans =
9 12 15
19 26 33
$A \backslash B$ is equivalent to $A^{-1} B$, and $A/B$ is equivalent to $A B^{-1}$
The inverse of a matrix can be obtained using the Octave function inv or pinv.
Octave has commands for generating special matrices. For example, you can create a diagonal matrix with the diag command using a vector containing the diagonal elements as the input argument, such as
%%octave
D = diag ([1 2])
D =
Diagonal Matrix
1 0
0 2
%%octave
A = [1 2; 3 4; 5 6]
v = [1 2 3]
v = [1; 2; 3]
v = [1:0.1:2]
v = 1:6
C = 2*ones(2,3)
w = ones(1,3)
w = zeros(1,3)
w = rand(1,3)
w = randn(1,3)
w = -6 + sqrt(10)*(randn(1,10000))
hist(w)
A =
1 2
3 4
5 6
v =
1 2 3
v =
1
2
3
v =
1.00000 1.10000 1.20000 1.30000 1.40000 1.50000 1.60000 1.70000 1.80000 1.90000 2.00000
v =
1 2 3 4 5 6
C =
2 2 2
2 2 2
w =
1 1 1
w =
0 0 0
w =
0.76678 0.80289 0.22341
w =
-0.41749 1.51084 0.84159
w =
1.0e+01 *
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Columns 9617 through 9632:
-0.27819 -0.37779 -0.44664 -0.93537 -0.72283 -0.57810 -0.78635 -0.37889 -0.17063 -0.61863 -0.52738 -0.13411 -0.95352 -0.46258 -0.50210 -0.03823
Columns 9633 through 9648:
-0.54025 -0.44018 -0.41457 -1.22123 -0.80896 -0.28212 -0.46975 -0.49035 -0.16788 -0.55227 -0.73634 -0.81445 -0.19364 -0.71481 0.11656 -0.59621
Columns 9649 through 9664:
-0.20908 -0.56345 -0.83042 -0.44403 -0.39673 -0.06538 -0.61698 -1.06885 -1.34350 -0.68263 -0.67873 -0.92056 -1.08834 -0.88920 -1.12794 -0.95905
Columns 9665 through 9680:
-0.88221 -0.70995 -0.13162 -0.16420 -0.69669 -0.64778 -0.34197 -0.38631 -0.22117 -0.66676 -1.22446 -0.40716 -0.29956 -0.83389 -0.39850 -0.99094
Columns 9681 through 9696:
-1.34667 -0.05245 -0.83913 -0.71255 -0.77556 -0.80641 -0.84027 -1.85043 -0.62864 -1.10363 -0.81241 -0.65698 -0.68796 -0.49724 -0.20270 -0.72542
Columns 9697 through 9712:
-0.16760 -0.61626 -0.38376 -0.50970 -0.76704 -0.24063 -0.93228 -0.29589 0.07874 -0.52810 -0.11647 -0.40265 -1.02030 -1.39647 -1.16330 -0.55413
Columns 9713 through 9728:
-0.32617 0.53500 -0.52753 -0.70267 -0.88369 -0.60017 -1.11489 -0.41859 -0.18717 -0.91587 0.00091 -0.90873 -0.77708 -0.43108 -0.26888 -0.56053
Columns 9729 through 9744:
-1.11732 -0.28666 0.12395 -0.38773 -0.47493 -0.59639 -0.79722 -0.47227 -0.31224 -0.99239 -1.25394 -0.97453 -0.54858 -0.32356 -0.53691 -0.33777
Columns 9745 through 9760:
-0.64883 -1.86235 -0.69049 -0.55883 -0.02461 -0.40905 -0.54992 -0.51222 -0.73663 -0.96007 -0.71309 -0.12156 -0.89057 -0.58033 -1.07521 -0.68252
Columns 9761 through 9776:
-0.34568 -0.26073 -0.27418 -0.43284 -0.74133 -1.04939 0.21791 -0.23329 -0.46915 -0.84885 -0.48662 -0.38959 -0.56270 -0.67477 -0.11698 -0.79405
Columns 9777 through 9792:
-0.15750 -0.96904 -0.56459 -1.10564 -0.41207 -0.58478 -0.09046 -0.34375 -0.90091 -0.58711 -0.69259 -1.07284 0.25246 -0.56788 -0.56248 -0.47499
Columns 9793 through 9808:
-0.85270 -0.80940 -0.59179 -1.42615 -0.46957 -0.56731 -0.67675 -0.67374 -0.33836 0.07088 -1.10074 -0.86282 -0.68169 -0.17375 -1.07303 -0.29991
Columns 9809 through 9824:
-0.43276 -0.71747 -0.87578 -0.57329 -0.34515 -0.56594 -0.57335 -0.71327 -0.52670 -0.64035 -1.18096 -0.46377 -0.56777 -0.82163 -0.83484 -0.64687
Columns 9825 through 9840:
-0.82847 -0.54163 -0.44455 -0.44529 -0.01538 -0.82691 -0.56330 -0.50763 -0.92507 -0.61332 0.35998 -0.10728 -0.80364 -1.06487 -0.62151 -0.75882
Columns 9841 through 9856:
-0.77627 -0.82082 -0.42724 -0.83094 -0.59019 -0.83153 -0.30956 -0.43427 -0.34029 -0.60309 -0.36863 -0.62885 -0.70826 -0.92321 -0.82605 -1.05534
Columns 9857 through 9872:
-0.79114 -0.28172 -0.85970 -0.68733 -0.45174 -0.64819 -0.51695 -0.70222 -0.40991 -0.84178 -0.75405 -0.94458 -0.83820 -0.73173 -0.21494 -0.62154
Columns 9873 through 9888:
-0.49179 -0.40028 -0.61018 0.10627 -0.37063 -0.80863 -0.36590 -0.26185 -0.54301 -0.36098 -0.51564 -0.32682 -1.02203 -1.33493 -0.70801 -0.45153
Columns 9889 through 9904:
-0.79446 -0.79952 -0.52185 -0.61221 -0.37820 -0.91183 -0.12814 -0.32964 -0.77958 -1.16460 -0.59314 -0.31158 -0.64429 -0.30694 -1.05205 -0.43984
Columns 9905 through 9920:
-0.74561 -0.84322 -0.60670 0.05544 -0.42479 -0.52600 -0.74991 -0.48881 -0.25925 -0.57810 -1.05458 -0.40939 -0.12136 -0.61936 -0.58237 -0.55489
Columns 9921 through 9936:
-0.35498 -0.65711 -0.48781 -0.49024 -0.55562 -0.43732 -0.84643 -0.03677 -0.18136 -1.34752 -0.68851 -0.50667 -0.35567 -0.57842 -0.68279 -0.94042
Columns 9937 through 9952:
-0.83140 -0.65619 -0.46082 -0.77789 -0.56941 -1.35415 -0.86807 -0.38256 -0.22960 -0.33913 -0.74613 -0.11451 -0.20312 -0.69880 -0.57953 -0.73296
Columns 9953 through 9968:
-0.32020 -0.93780 -0.79800 -0.44721 -0.89170 -0.72026 -0.37568 -0.67779 -0.35766 -0.14850 -0.36878 -0.55012 -0.34658 -1.12584 -0.87873 -0.89058
Columns 9969 through 9984:
-0.71022 -0.53366 -0.83969 -0.63530 -0.50314 -0.09252 -0.45137 -0.77297 -0.08257 -1.16470 -0.52990 -0.82941 -0.66950 -0.69281 -0.79761 -0.85440
Columns 9985 through 10000:
-0.94543 -0.23804 -0.76867 -1.01520 -0.34632 -0.49287 -0.25078 -0.64224 -0.22581 -0.68020 -0.60789 -0.74147 -0.62553 -1.02389 -0.95065 -1.37070
2.6 Octave Special Utility Matrices
from IPython.display import IFrame
IFrame('https://www.gnu.org/software/octave/doc/interpreter/Special-Utility-Matrices.html', width='100%', height=540)
<iframe
width="100%"
height="540"
src="https://www.gnu.org/software/octave/doc/interpreter/Special-Utility-Matrices.html"
frameborder="0"
allowfullscreen
></iframe>
%%octave
I = eye(5) % 5x5 identity matrix
I =
Diagonal Matrix
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
2.7 Basic Matrix Functions
from IPython.display import IFrame
IFrame('https://www.gnu.org/software/octave/doc/interpreter/Basic-Matrix-Functions.html', width='100%', height=540)
<iframe
width="100%"
height="540"
src="https://www.gnu.org/software/octave/doc/interpreter/Basic-Matrix-Functions.html"
frameborder="0"
allowfullscreen
></iframe>
2.8 Vector Operation
An n vector is a row vector or a column array of n numbers. In Octave, elements enclosed by brackets and separated by semicolon generate a column vector.
The transpose of a row vector is a column vector, and vice versa. This can be done in Octave using the symbol (‘).
%%octave
A = [1 2; 3 4]
A'
A =
1 2
3 4
ans =
1 3
2 4
Vectors of the same size can be added or subtracted. The operation ( .* ) performs element-by-element multiplication.
The (:) can also be used to generate a row vector.
%%octave
x = 1:8
x =
1 2 3 4 5 6 7 8
For increment other than unity, the following command can be used:
%%octave
x = 0:pi/3:5*pi
x =
1.0e+01 *
Columns 1 through 15:
0.00000 0.10472 0.20944 0.31416 0.41888 0.52360 0.62832 0.73304 0.83776 0.94248 1.04720 1.15192 1.25664 1.36136 1.46608
Column 16:
1.57080
For negative increment
%%octave
x = 5 : -1 : -4
x =
5 4 3 2 1 0 -1 -2 -3 -4
2.9 Elementary Function
Some of the Octave function automatically operate element by element on an array. For example, exp(x) will return an array with each element equal to the exponential of the corresponding element of x.
%%octave
abs(-10) % Absolute value
acos(0.5) % Inverse cosine
angle(1+j) % Phase angle
asin(0.5) % Inverse sine
atan(1) % Inverse tangent
conj(1 - j) % complex conjugate
cos(30) % Cosine
exp(-0.5) % Exponential
fix(0.8) % Round towards zero
floor(1.8) % Round towards infinity
imag(1+j) % Complex Imaginary part
log(e) % Natural logarithm
log10(100) % Common logarithm
real(1+j) % Complex real part
rem(10,8) % remainder after division
round(1.5) % Round towards nearest integer
sign(0) % Signum function
sqrt(4) % Square root
tan(pi/4) % Tangent
ans = 10
ans = 1.0472
ans = 0.78540
ans = 0.52360
ans = 0.78540
ans = 1 + 1i
ans = 0.15425
ans = 0.60653
ans = 0
ans = 1
ans = 1
ans = 1
ans = 2
ans = 1
ans = 2
ans = 2
ans = 0
ans = 2
ans = 1.00000
2.10 Logical Operators
Octave’s relational operators and logical operators also work on an element-by-element basis. Relational operators compare two scalars and produce a 1 if the operation is true and a 0 if it is false. For example, if you enter t = 17 > 55, Octave will respond with t = 0. When used with two matrices, relational operators compare corresponding matrix elements. For example, L = D < = X will check every element of D against the corresponding element of X. If the element of D is less than or equal to the corresponding element of X, the corresponding element of L will be 1. Otherwise, the corresponding element of L will be zero.
The logical operators & for logical AND, | for logical OR, and ~ for logical NOT all return 1 for logical TRUE and 0 for logical FALSE.
%%octave
1 == 2
1 ~= 2
1 && 0
1 || 0
xor(1,0)
ans = 0
ans = 1
ans = 0
ans = 1
ans = 1
2.11 Evaluate m-files
A script file is an ASCII file that contains a series of Octave command just as you would enter them in Octave environment. Statement that begins with % is considered to be comment and are ignored by Octave. The script file is created outside the Octave environment with any text editor or word processor. Each script file should have a name that ends in “.m” The commands in the script file are executed in the Octave environment by simply entering the name of the script file without the extension “.m”
%ls
[0m[01;35mCostContourPlot.png[0m ML0.pdf ml2y.dat
CostFunctionContour.m ML0.snm ML3.ipynb
CostFunction.m ML0.tex ml3x.dat
[01;35mCostSurfacePlot.png[0m ML0.toc ml3y.dat
GradientDescentLines.m ML0.vrb mydft.m
GradientDescent.m [01;35mml1data.png[0m myfactorial.m
[01;34mImages[0m/ ML1.ipynb OctaveFundamentals.ipynb
Makefile [01;35mml1prediction.png[0m octave-workspace
ML0.aux ml1x.dat plotData.m
ML0.log ml1y.dat Prediction.m
ML0.nav ML2.ipynb rms.m
ML0.out ml2x.dat template.tex
%cat plotData.m
% Based on Andrew Ng's Machine Learning course
clc
x = load('ml1x.dat');
y = load('ml1y.dat');
m = length(x);
figure, plot(x,y,'rh', 'markerfacecolor', 'auto');
xlabel('Age (yrs.)');
ylabel('Height (m)');
print -dpng ml1data
%%octave -f svg -s 720,480
eval("plotData");
[H[2JGPL Ghostscript 9.05: Error: Font Renderer Plugin ( FreeType ) return code = -1
GPL Ghostscript 9.05: Error: Font Renderer Plugin ( FreeType ) return code = -1
GPL Ghostscript 9.05: Error: Font Renderer Plugin ( FreeType ) return code = -1
GPL Ghostscript 9.05: Error: Font Renderer Plugin ( FreeType ) return code = -1
GPL Ghostscript 9.05: Error: Font Renderer Plugin ( FreeType ) return code = -1
GPL Ghostscript 9.05: Error: Font Renderer Plugin ( FreeType ) return code = -1
GPL Ghostscript 9.05: Error: Font Renderer Plugin ( FreeType ) return code = -1
GPL Ghostscript 9.05: Error: Font Renderer Plugin ( FreeType ) return code = -1
2.12 Getting Help
from oct2py import octave
help(octave.ones)
Help on function ones in module oct2py.core:
ones(*args, **kwargs)
`ones' is a built-in function
-- Built-in Function: ones (N)
-- Built-in Function: ones (M, N)
-- Built-in Function: ones (M, N, K, ...)
-- Built-in Function: ones ([M N ...])
-- Built-in Function: ones (..., CLASS)
Return a matrix or N-dimensional array whose elements are all 1.
If invoked with a single scalar integer argument N, return a square
NxN matrix. If invoked with two or more scalar integer arguments,
or a vector of integer values, return an array with the given
dimensions.
If you need to create a matrix whose values are all the same, you
should use an expression like
val_matrix = val * ones (m, n)
The optional argument CLASS specifies the class of the return array
and defaults to double. For example:
val = ones (m,n, "uint8")
See also: zeros
Additional help for built-in functions and operators is
available in the on-line version of the manual. Use the command
`doc <topic>' to search the manual index.
Help and information about Octave is also available on the WWW
at http://www.octave.org and via the help@octave.org
mailing list.
Oct2Py Parameters
--------------------------
inputs : array_like
Variables to pass to the function.
verbose : bool, optional
Log Octave output at INFO level. If False, log at DEBUG level.
nout : int, optional
Number of output arguments.
This is set automatically based on the number of return values
requested.
You can override this behavior by passing a different value.
timeout : float, optional
Time to wait for response from Octave (per character).
plot_dir: str, optional
If specificed, save the session's plot figures to the plot
directory instead of displaying the plot window.
plot_name : str, optional
Saved plots will start with `plot_name` and
end with "_%%.xxx' where %% is the plot number and
xxx is the `plot_format`.
plot_format: str, optional
The format in which to save the plot.
plot_width: int, optional
The plot with in pixels.
plot_height: int, optional
The plot height in pixels.
kwargs : dictionary, optional
Key - value pairs to be passed as prop - value inputs to the
function. The values must be strings or numbers.
Returns
-----------
out : value
Value returned by the function.
Raises
----------
Oct2PyError
If the function call is unsucessful.
Notes
-----
Integer type arguments will be converted to floating point
unless `convert_to_float=False`.
2.13 Creating Functions
Function files are m-files that are very similar to script files. The first major difference is that the first line of a function file begins with the word function, followed by a statement identifying the name of the function and the input and output argument in the form.
function [output arguments] = function_name(input arguments)
For example, suppose you want to create a new function called rms that computes the root mean square of a list of numbers. The first line of your function file might look like function y = rms(v). The next several lines should be comment line describing how to use rms. Later when you type help rms, Octave will respond with these comment lines. The remaining line of the function file should look as they would in a script file. Remember that the input argument v will be defined when the function is called.
The second major difference between function files and script files is that the variables generated in function files are local to the function, whereas variables in script files are global. The function file to produce the rms command might look as follows:
function y = rms(v)
% rms Root mean square
% rms(v) returns the root mean square of the elements
% of column vector v. If v is a matrix then
% rms(v) returns a row vector such that each element
% is the root mean square of the elements in the corresponding
% column of v.
vs = v.^2;
s = size(v);
y = sqrt(sum(vs)/s(1));
%cat rms.m
function y = rms(v)
% rms Root mean square
% rms(v) returns the root mean square of the elements
% of column vector v. If v is a matrix then
% rms(v) returns a row vector such that each element
% is the root mean square of the elements in the corresponding
% column of v.
vs = v.^2;
s = size(v);
y = sqrt(sum(vs)/s(1));
%%octave
help("rms")
`rms' is a function from the file /home/cobalt/repos/mlintroduction/rms.m
rms Root mean square
rms(v) returns the root mean square of the elements
of column vector v. If v is a matrix then
rms(v) returns a row vector such that each element
is the root mean square of the elements in the corresponding
column of v.
Additional help for built-in functions and operators is
available in the on-line version of the manual. Use the command
`doc <topic>' to search the manual index.
Help and information about Octave is also available on the WWW
at http://www.octave.org and via the help@octave.org
mailing list.
%%octave
v = [ 1; 2; 3]
rms(v)
v =
1
2
3
ans = 2.1602
2.14 Graphics
Basic plotting
%%octave
t = [0:0.01:0.98];
y1 = sin(2*pi*4*t);
plot(t,y1);
%%octave
t = [0:0.01:0.98];
y1 = sin(2*pi*4*t);
plot(t,y1);
y2 = cos(2*pi*4*t);
hold on;
plot(t,y2,'r');
xlabel('time');
ylabel('value');
legend('sin','cos');
title('my plot');
%%octave -f PNG
t = [0:0.01:0.98];
y1 = sin(2*pi*4*t);
y2 = cos(2*pi*4*t);
subplot(1,2,1);
plot(t,y1);
subplot(1,2,2);
plot(t,y2);
axis([0 1 -1 1]);
III. Laboratory Exercise
3.1 Encode the following matrices in MATLAB.
%%octave
A = [2 5 -6; 4 -4 3; 1 3 -5]
B = [1 5 -6; 4 -2 3; 2 3 5]
AB = A*B
BA = B*A
SUM = A+B
DIFF = A-B
Ainv = inv(A)
Atrans = A'
Axinv = A/A
A =
2 5 -6
4 -4 3
1 3 -5
B =
1 5 -6
4 -2 3
2 3 5
AB =
10 -18 -27
-6 37 -21
3 -16 -22
BA =
16 -33 39
3 37 -45
21 13 -28
SUM =
3 10 -12
8 -6 6
3 6 0
DIFF =
1 0 0
0 -2 0
-1 0 -10
Ainv =
0.26829 0.17073 -0.21951
0.56098 -0.09756 -0.73171
0.39024 -0.02439 -0.68293
Atrans =
2 4 1
5 -4 3
-6 3 -5
Axinv =
1.00000 -0.00000 -0.00000
-0.00000 1.00000 -0.00000
-0.00000 0.00000 1.00000
%%octave
A = [2 5 -6; 4 -4 3; 1 3 -5]
B = [1 5 -6; 4 -2 3; 2 3 5]
AxB_elementwise = A.*B
AdivB_elementwise = A./B
A(:,1)'.*B(2,:) % Column 1 of Matrix A times Row 2 of Matrix B
A =
2 5 -6
4 -4 3
1 3 -5
B =
1 5 -6
4 -2 3
2 3 5
AxB_elementwise =
2 25 36
16 8 9
2 9 -25
AdivB_elementwise =
2.00000 1.00000 1.00000
1.00000 2.00000 1.00000
0.50000 1.00000 -1.00000
ans =
8 -8 3
Generate a row vector n whose value is linearly increasing from 0 to 127.
%%octave
v = 0:127
y = sin(2*pi*v/25);
subplot(2,1,1)
plot(v,y, 'linewidth',3)
xlim([0 127])
xlabel('t')
ylabel('sin(t)')
title('Continuous Plot');
grid on;
subplot(2,1,2)
stem(v,y, 'linewidth',3)
xlim([0 127])
xlabel('n')
ylabel('x[n]')
title('Discrete Plot');
v =
Columns 1 through 25:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Columns 26 through 50:
25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
Columns 51 through 75:
50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74
Columns 76 through 100:
75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
Columns 101 through 125:
100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124
Columns 126 through 128:
125 126 127
Given a vector x whose values are complex, determine the magnitude and the corresponding phase. Use the Octave command angle and abs. To ensure that the angle is within -π to π range, use the Octave function unwrap. Plot the magnitude and phase with corresponding label. Use the function stem to plot the magnitude and phase. The two graph should be shown in one screen. (Discrete plot - “stem”)
• abs – Computes the magnitude of a complex number.
• angle – Computes the phase angle of a complex number.
• stem – Draws discrete plots.
%%octave
x = [1-j 2+j 4+ 8j -1-i]
r = abs(x)
theta = angle(x)
thetauw = unwrap(theta)
figure, stem(r,'linewidth', 5, 'fill')
ylim([0 10])
xlim([0 5])
xlabel('n')
ylabel('Magnitude')
title('Amplitude plot')
grid on
figure, stem(theta,'linewidth',5, 'fill')
ylim([-pi pi])
xlim([0 5])
xlabel('n')
ylabel('Angle')
title('Phase plot')
grid on
figure, stem(thetauw,'linewidth',5, 'fill')
ylim([-pi 2*pi])
xlim([0 5])
xlabel('n')
ylabel('Angle')
title('Phase plot (w/ unwrap)')
grid on
x =
1 - 1i 2 + 1i 4 + 8i -1 - 1i
r =
1.41421 2.23607 8.94427 1.41421
theta =
-0.78540 0.46365 1.10715 -2.35619
thetauw =
-0.78540 0.46365 1.10715 3.92699
%%octave
help unwrap
`unwrap' is a function from the file /usr/share/octave/3.6.2/m/signal/unwrap.m
-- Function File: B = unwrap (X)
-- Function File: B = unwrap (X, TOL)
-- Function File: B = unwrap (X, TOL, DIM)
Unwrap radian phases by adding multiples of 2*pi as appropriate to
remove jumps greater than TOL. TOL defaults to pi.
Unwrap will work along the dimension DIM. If DIM is unspecified
it defaults to the first non-singleton dimension.
Additional help for built-in functions and operators is
available in the on-line version of the manual. Use the command
`doc <topic>' to search the manual index.
Help and information about Octave is also available on the WWW
at http://www.octave.org and via the help@octave.org
mailing list.
Using for loop, create a function that will calculate factorial of x, where the element of x is any scalar integer. Call this function myfactorial.
%cat myfactorial.m
function result = myfactorial(n)
if( n == 0 )
result = 1;
return;
else
result = 1;
for i = 1:n
result = result*i;
endfor
endif
endfunction
%%octave
eval('myfactorial(5)')
eval('myfactorial(4)')
eval('myfactorial(3)')
eval('myfactorial(2)')
eval('myfactorial(1)')
eval('myfactorial(0)')
ans = 120
ans = 24
ans = 6
ans = 2
ans = 1
ans = 1
where $W = e^{-j2\pi / N}$
%cat mydft.m
function X = mydft(x)
N = length(x);
X = zeros(N,1);
for k = 0:N-1
for n = 0:N-1
X(k+1) = X(k+1) + x(n+1)*exp(-j*2*n*k*pi/N);
endfor
endfor
% Set-up plot domain
t = 0:N-1;
%subplot(212)
figure, stem(t,angle(X),'m--');
xlabel('Frequency');
ylabel('Phase');
title('Frequency domain - Phase response')
grid on;
%subplot(211)
figure, stem(t,abs(X),'r--');
set(gca,'yscale','log');
xlabel('Frequency');
ylabel('|X(k)|');
title('Frequency domain - Magnitude response')
grid on;
endfunction
Sinc Input
%%octave
x = sinc([-5:1/10:5]);
t = 0:length(x)-1;
stem(t,x,'fill');
xlabel('Time (s)');
ylabel('Amplitude');
title('Time domain - Input sequence');
grid on;
DFT using user function
%%octave
x = sinc([-5:1/10:5]);
format short g
eval("mydft(x)");
warning: axis: omitting non-positive data in log plot
ans =
10.399 + 0i
-9.5807 - 0.2981i
10.451 + 0.65097i
-9.3531 - 0.87532i
10.91 + 1.3645i
-5.3327 - 0.83611i
-0.77582 - 0.1465i
-0.3853 - 0.085244i
-0.24054 - 0.061123i
-0.16731 - 0.0481i
-0.12398 - 0.039857i
-0.09578 - 0.034113i
-0.076208 - 0.029844i
-0.061978 - 0.026523i
-0.051262 - 0.02385i
-0.042965 - 0.02164i
-0.036397 - 0.019774i
-0.031099 - 0.018171i
-0.026761 - 0.016774i
-0.023161 - 0.015541i
-0.020139 - 0.014441i
-0.017578 - 0.013452i
-0.015389 - 0.012555i
-0.013504 - 0.011735i
-0.011869 - 0.010981i
-0.010444 - 0.010283i
-0.0091947 - 0.009634i
-0.0080947 - 0.0090276i
-0.0071222 - 0.0084582i
-0.0062597 - 0.0079214i
-0.0054925 - 0.0074133i
-0.0048084 - 0.0069306i
-0.0041972 - 0.0064704i
-0.0036503 - 0.0060303i
-0.0031607 - 0.0056079i
-0.002722 - 0.0052015i
-0.0023292 - 0.0048093i
-0.0019778 - 0.0044298i
-0.0016639 - 0.0040615i
-0.0013842 - 0.0037033i
-0.0011361 - 0.003354i
-0.00091705 - 0.0030125i
-0.00072501 - 0.002678i
-0.00055826 - 0.0023495i
-0.00041532 - 0.0020261i
-0.00029494 - 0.0017072i
-0.00019611 - 0.0013919i
-0.000118 - 0.0010796i
-5.9961e-05 - 0.00076952i
-2.1528e-05 - 0.00046107i
-2.3888e-06 - 0.00015359i
-2.3888e-06 + 0.00015359i
-2.1528e-05 + 0.00046107i
-5.9961e-05 + 0.00076952i
-0.000118 + 0.0010796i
-0.00019611 + 0.0013919i
-0.00029494 + 0.0017072i
-0.00041532 + 0.0020261i
-0.00055826 + 0.0023495i
-0.00072501 + 0.002678i
-0.00091705 + 0.0030125i
-0.0011361 + 0.003354i
-0.0013842 + 0.0037033i
-0.0016639 + 0.0040615i
-0.0019778 + 0.0044298i
-0.0023292 + 0.0048093i
-0.002722 + 0.0052015i
-0.0031607 + 0.0056079i
-0.0036503 + 0.0060303i
-0.0041972 + 0.0064704i
-0.0048084 + 0.0069306i
-0.0054925 + 0.0074133i
-0.0062597 + 0.0079214i
-0.0071222 + 0.0084582i
-0.0080947 + 0.0090276i
-0.0091947 + 0.009634i
-0.010444 + 0.010283i
-0.011869 + 0.010981i
-0.013504 + 0.011735i
-0.015389 + 0.012555i
-0.017578 + 0.013452i
-0.020139 + 0.014441i
-0.023161 + 0.015541i
-0.026761 + 0.016774i
-0.031099 + 0.018171i
-0.036397 + 0.019774i
-0.042965 + 0.02164i
-0.051262 + 0.02385i
-0.061978 + 0.026523i
-0.076208 + 0.029844i
-0.09578 + 0.034113i
-0.12398 + 0.039857i
-0.16731 + 0.0481i
-0.24054 + 0.061123i
-0.3853 + 0.085244i
-0.77582 + 0.1465i
-5.3327 + 0.83611i
10.91 - 1.3645i
-9.3531 + 0.87532i
10.451 - 0.65097i
-9.5807 + 0.2981i
Checking using the built-in FFT function
%%octave -f png
x = sinc([-5:1/10:5]);
t = 0:length(x)-1;
%subplot(212)
figure, stem(t,angle(fft(x)),'m--');
xlabel('Frequency');
ylabel('Phase');
title('Frequency domain - Phase response')
grid on;
%subplot(211)
figure, stem(t,abs(fft(x)),'r--');
set(gca,'yscale','log');
xlabel('Frequency');
ylabel('|X(k)|');
title('Frequency domain - Magnitude response')
grid on;
warning: axis: omitting non-positive data in log plot
– mkc
Written on September 19, 2015