-seminar 1 for digital signal processing Using Matlab -seminar 1 for digital signal processing

Matlab as calculator 1. basic arithmetic operator + - * / ^ () e.g. 2+3/4*5 = 3^2*4 = 3-4/4-2 = 2 extended arithmetic - accidental error 1/0 = -1/0 = 0/0 = 1/Inf = we know 1/(1/x) = x, then try 1/(1/0) = ? complex number 1+i , -1+3*i

Numbers and formats 1. Different kind of numbers Integer: e.g. 123, -218 Real : e.g. 1.234, - 10.9 complex : e.g. 3.21-3.4*i (i = sqrt(-1)). Inf : Infinity (dividing by 0) NaN: Not a number (0/0) e: notation for very large or small number, e.g. -1.34e+03 = ? , 1.34e-05 = ?

Numbers and formats 2. Calculation: 15 significant figures The 'format' tells how matlab prints numbers. Type 'help format' in command window for full list e.g. pi = ?? usually 3.1416 format long pi = ?? format short e format short If want to switch back to default format, type: format

Numbers and formats 3. finite accuracy consequences Matlab limit accuracy (enough for most cases): 64 bits, store number as large as 2*10^308, as small as 2*10^(-308) Store any number 15 significant figures: e.g. 1.23456789023456 (14 figures, can handle) 1.23456789023456789012 (20 digits, truncated to 15 figures) round off cannnot be avoid. e.g. what is sin (pi) = ? try sin(pi) = ??-slight round-off error, take it as zero as long as small like 10^(-15).

Variables 1. combination of letter and number, case sensitive e.g. a , x1, z2453, A, t = 2+3-9, 2*t-2 Not allowed: Net-c, 2p, %x, @sign 2. special names: eps (= 2^(-54)), pi --> avoid using 3. complex numbers : i, j = sqrt(-1), unless you change them

Suppressing output (don't want to show output) hidden: x = -13; (semi-colon).

Build-in function sin, cos, tan, sec = 1/sin, cosec = 1/cos, cotan e.g. work out the coordinate of a point on a circle of radius 5 centred at origin, having an elevation 30 degree = pi/6 radians. 2. inverse trig function e.g. asin, acos, atan--> answer returned in radians, so asin(1) = pi/2 3. exponential exp : exp(x) = e^x logarithm: log: log to base e/ log10 to base 10 square root: sqrt(). e.g. x = 9; sqrt(x), exp(x), log(sqrt(x)), log10(x^2+6)

vectors 1. row vectors a = [1 2 3] or a = [1, 2, 3] e.g. V = [1 3 sqrt(5)], what is length(V) - space vitally important : e.g. v2 = [3+ 4 5], v3 = [3 +4 5]; - add vector of the same length: e.g. V + v3, v4 = 3*v3, v5 = 2*V-3*v4, v6 = v+v2??? wrong! since dimension must agree build a row vector from existing ones: e.g. w = [1 2 3], z = [8, 9], cd = [2*z -w], sort(cd) (ascending order) look at value of particular entries: e.g. w(2) = ? - set w(3) = 100, then w = ??

Vectors 2. column vector e.g. c = [1; 3; sqrt(5)] or c2 = [3 return 4 return 5] c3 = 2*c-5*c2 3. column notation : a shortcut for producing row vectors e.g. 1:100 3:7 5:0.1:6 1:-1 --> [] 0.32:0.1:0.6 -0.4:-0.3:-2

vector operation 1. scalar product: u*v = sum (ui*vi) u = [u1, ..., un]; v = [v1;...; vn] e.g. u = [10 -11 12], v = [20; -21; -22]; prod = u*v e.g. w = [2 1 3]; z = [7; 6; 5]; check: v*w, u*w', u*u', v'*z - norm of a vector: ||u|| = sqrt(sum(ui)) compute norm: sqrt(u*u') or norm(u)

vector operation 2. dot product-vector of the same length times with each other u.v = [u1v1,...,unvn] e.g. u.*w, u.*v', u.*z, u'.*v ex.: Tabulate y = x*sin(pi*x) for x = 0, 0.25, ... , 1 ans: x = 0:.25:1; y = x.*sin(pi*x);

Vector operations 3. dot divison of array-element by element division e.g. a = 1:5, b = 6:10, check a./b = , a./a = , c = -2:2, a./c, a.*b-24, ans./c - ex: limit sin(pi*x)/x, as x-->0 - ans: x = [.1 .01. .001 .0001], sin(pi*x)./x, format long , ans - pi - e.g. 1/x (wrong!), 1./x (correct) 4. dot power of array (.^) sqare all element of a vector e.g. u.*u, u.^2, u.^4, u.*w.^(-2)

Entering matrices Defining a matrix in ML is simple: just list its entries between [ ]. Use semicolons to separate rows. A= [1 2 3; 4 5 6; 7 8 9] A = 1 2 3 4 5 6 7 8 9

Addressing matrix elements A(2, 3) a single matrix element A(2:3, 1:2) a portion of the matrix ans = 4 5 7 8

Manipulating matrices - 1 You can sum matrices of same size A= [1 2; 4 5]; B = A’; C = 0.5 * (A + B) C = 1 3 3 5 NB: The apostrophe (‘) indicates transpose operation, i.e. A’ is the transpose of A.

Manipulating matrices - 2 You can multiply a matrix by a scalar A= [1 2; 4 5]; k = pi; B = k*A B = 3.1416 6.2832 12.5664 15.7080 All the matrix elements get multiplied by k.

Manipulating matrices - 3 Matrix can be multiplied together (provided ...) A= rand(5,2), B = rand(2, 5), C = A * B Likewise you can take powers of a (square) matrix, S_1 = rand(5,5), S_3 = S^3 NB: The function rand(n,m) returns an nxm matrix of random numbers uniformly distributed between 0 and 1.

Manipulating matrices - 4 Matrices can be multiplied ‘element-by-element’ using the dot notation for multiplication A = [1 2; 3 4], B = [10 20; 30 40], C = A .* B C = 10 40 90 160 Likewise we have a dot-notation for powers, e.g. D = A.^5

Useful matrices All matrices below are nxm A = zeros(n,m) all zeros A = ones(n,m) all ones A = rand(n,m) uniformely distributed [0,1] A = randn(n, m) normally distributed (mean=0, SD=1)

Plotting Use plot(x, y) to plot vector y vs. the vector x. NB: x and y should have the same size Example t = 0:0.001:0.6; x = sin(2*pi*50*t)+sin(2*pi*120*t); y = x + 2*randn(size(t)); plot(1000*t(1:50),y(1:50)) title('Signal Corrupted with Zero-Mean Random Noise') xlabel('time (milliseconds)')

Plotting power spectra Y = fft(y,512); Pyy = Y.* conj(Y) / 512; f = 1000*(0:256)/512; plot(f,Pyy(1:257)) title('Frequency content of y') xlabel('frequency (Hz)')

Scripts Scripts have no input or output arguments. They're useful for automating series of ML commands. Scripts operate on existing data in the workspace, or they can create new data on which to operate. To create a script: File->New->M-file Invoke a script by typing its name from the prompt.

Functions Functions are M-files that accept input arguments and return output arguments. They operate on variables within their own workspace. See an example below function y = average(x) % AVERAGE Mean of vector elements. % AVERAGE(X), where X is a vector, is the mean of vector elements. [m,n] = size(x); if (~((m == 1) | (n == 1)) | (m == 1 & n == 1)) error('Input must be a vector') end y = sum(x)/length(x); % Actual computation

Exercise: build a DTMF (touch-tone) dialing pad Freq (Hz) 1209 1336 1477 1633 697 1 2 3 A 770 4 5 6 B 852 7 8 9 C 941 * # D NB: Get info about function ‘sound’