This appendix provides Matlab and Octave examples for various topics covered in this book. The term `matlab' (uncapitalized) refers to either Matlab or Octave .
Complex Numbers in Matlab and Octave
Matlab and Octave have the following primitives for complex numbers:
octave:1> help j j is a built-in constant - Built-in Variable: I - Built-in Variable: J - Built-in Variable: i - Built-in Variable: j A pure imaginary number, defined as `sqrt (-1)'. The `I' and `J' forms are true constants, and cannot be modified. The `i' and `j' forms are like ordinary variables, and may be used for other purposes. However, unlike other variables, they once again assume their special predefined values if they are cleared *Note Status of Variables::. Additional help for built-in functions, operators, and variables is available in the on-line version of the manual. Use the command `help -i <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 firstname.lastname@example.org mailing list. octave:2> sqrt(-1) ans = 0 + 1i octave:3> help real real is a built-in mapper function - Mapping Function: real (Z) Return the real part of Z. See also: imag and conj. ... octave:4> help imag imag is a built-in mapper function - Mapping Function: imag (Z) Return the imaginary part of Z as a real number. See also: real and conj. ... octave:5> help conj conj is a built-in mapper function - Mapping Function: conj (Z) Return the complex conjugate of Z, defined as `conj (Z)' = X - IY. See also: real and imag. ... octave:6> help abs abs is a built-in mapper function - Mapping Function: abs (Z) Compute the magnitude of Z, defined as |Z| = `sqrt (x^2 + y^2)'. For example, abs (3 + 4i) => 5 ...
octave:7> help angle angle is a built-in mapper function - Mapping Function: angle (Z) See arg. ...Note how helpful the ``See also'' information is in Octave (and similarly in Matlab).
Complex Number Manipulation
Let's run through a few elementary manipulations of complex numbers in Matlab:
>> x = 1; >> y = 2; >> z = x + j * y z = 1 + 2i >> 1/z ans = 0.2 - 0.4i >> z^2 ans = -3 + 4i >> conj(z) ans = 1 - 2i >> z*conj(z) ans = 5 >> abs(z)^2 ans = 5 >> norm(z)^2 ans = 5 >> angle(z) ans = 1.1071
Now let's do polar form:
>> r = abs(z) r = 2.2361 >> theta = angle(z) theta = 1.1071
Curiously, is not defined by default in Matlab (though it is in
Octave). It can easily be computed in Matlab as
Below are some examples involving imaginary exponentials:
>> r * exp(j * theta) ans = 1 + 2i >> z z = 1 + 2i >> z/abs(z) ans = 0.4472 + 0.8944i >> exp(j*theta) ans = 0.4472 + 0.8944i >> z/conj(z) ans = -0.6 + 0.8i >> exp(2*j*theta) ans = -0.6 + 0.8i >> imag(log(z/abs(z))) ans = 1.1071 >> theta theta = 1.1071 >>Here are some manipulations involving two complex numbers:
>> x1 = 1; >> x2 = 2; >> y1 = 3; >> y2 = 4; >> z1 = x1 + j * y1; >> z2 = x2 + j * y2; >> z1 z1 = 1 + 3i >> z2 z2 = 2 + 4i >> z1*z2 ans = -10 +10i >> z1/z2 ans = 0.7 + 0.1i
Another thing to note about matlab syntax is that the transpose operator ' (for vectors and matrices) conjugates as well as transposes. Use .' to transpose without conjugation:
>>x = [1:4]*j x = 0 + 1i 0 + 2i 0 + 3i 0 + 4i >> x' ans = 0 - 1i 0 - 2i 0 - 3i 0 - 4i >> x.' ans = 0 + 1i 0 + 2i 0 + 3i 0 + 4i
Factoring Polynomials in Matlab
Let's find all roots of the polynomial
>> % polynomial = array of coefficients in matlab: >> p = [1 0 0 0 5 7]; % p(x) = x^5 + 5*x + 7 >> format long; % print double-precision >> roots(p) % print out the roots of p(x) ans = 1.30051917307206 + 1.10944723819596i 1.30051917307206 - 1.10944723819596i -0.75504792501755 + 1.27501061923774i -0.75504792501755 - 1.27501061923774i -1.09094249610903
Geometric Signal Theory
This section follows Chapter 5 of the text.
Vector Interpretation of Complex Numbers
Here's how Fig.5.1 may be generated in matlab:
>> x = [2 3]; % coordinates of x >> origin = [0 0]; % coordinates of the origin >> xcoords = [origin(1) x(1)]; % plot() expects coordinates >> ycoords = [origin(2) x(2)]; >> plot(xcoords,ycoords); % Draw a line from origin to x
The mean of a signal stored in a matlab row- or column-vector x can be computed in matlab as
mu = sum(x)/Nor by using the built-in function mean(). If x is a 2D matrix containing N elements, then we need mu = sum(sum(x))/N or mu = mean(mean(x)), since sum computes a sum along ``dimension 1'' (which is along columns for matrices), and mean is implemented in terms of sum. For 3D matrices, mu = mean(mean(mean(x))), etc. For a higher dimensional matrices x, ``flattening'' it into a long column-vector x(:) is the more concise form:
N = prod(size(x)) mu = sum(x(:))/Nor
mu = x(:).' * ones(N,1)/NThe above constructs work whether x is a row-vector, column-vector, or matrix, because x(:) returns a concatenation of all columns of x into one long column-vector. Note the use of .' to obtain non-conjugating vector transposition in the second form. N = prod(size(x)) is the number of elements of x. If x is a row- or column-vector, then length(x) gives the number of elements. For matrices, length() returns the greater of the number of rows or columns.I.1
Signal Energy and Power
In a similar way, we can compute the signal energy (sum of squared moduli) using any of the following constructs:
Ex = x(:)' * x(:) Ex = sum(conj(x(:)) .* x(:)) Ex = sum(abs(x(:)).^2)The average power (energy per sample) is similarly Px = Ex/N. The norm is similarly xL2 = sqrt(Ex) (same result as xL2 = norm(x)). The norm is given by xL1 = sum(abs(x)) or by xL1 = norm(x,1). The infinity-norm (Chebyshev norm) is computed as xLInf = max(abs(x)) or xLInf = norm(x,Inf). In general, norm is computed by norm(x,p), with p=2 being the default case.
The inner product of two column-vectors x and y (§5.9) is conveniently computed in matlab as
xdoty = y' * x
For real vectors x and y having the same length, we may compute the vector cosine by
cosxy = y' * x / ( norm(x) * norm(y) );For complex vectors, a good measure of orthogonality is the modulus of the vector cosine:
collinearity = abs(y' * x) / ( norm(x) * norm(y) );Thus, when collinearity is near 0, the vectors x and y are substantially orthogonal. When collinearity is close to 1, they are nearly collinear.
As discussed in §5.9.9, the orthogonal projection of onto is defined by
yx = (x' * y) * (x' * x)^(-1) * xMore generally, a length-N column-vector y can be projected onto the -dimensional subspace spanned by the columns of the N M matrix X:
yX = X * (X' * X)^(-1) * X' * yOrthogonal projection, like any finite-dimensional linear operator, can be represented by a matrix. In this case, the matrix
PX = X * (X' * X)^(-1) * X'is called the projection matrix.I.2Subspace projection is an example in which the power of matrix linear algebra notation is evident.
Projection Example 1
>> X = [[1;2;3],[1;0;1]] X = 1 1 2 0 3 1 >> PX = X * (X' * X)^(-1) * X' PX = 0.66667 -0.33333 0.33333 -0.33333 0.66667 0.33333 0.33333 0.33333 0.66667 >> y = [2;4;6] y = 2 4 6 >> yX = PX * y yX = 2.0000 4.0000 6.0000
Since y in this example already lies in the column-space of X, orthogonal projection onto that space has no effect.
Projection Example 2
Let X and PX be defined as Example 1, but now let
>> y = [1;-1;1] y = 1 -1 1 >> yX = PX * y yX = 1.33333 -0.66667 0.66667 >> yX' * (y-yX) ans = -7.0316e-16 >> eps ans = 2.2204e-16
In the last step above, we verified that the projection yX is orthogonal to the ``projection error'' y-yX, at least to machine precision. The eps variable holds ``machine epsilon'' which is the numerical distance between and the next representable number in double-precision floating point.
Orthogonal Basis Computation
Matlab and Octave have a function orth() which will compute an orthonormal basis for a space given any set of vectors which span the space. In Matlab, e.g., we have the following help info:
>> help orth ORTH Orthogonalization. Q = orth(A) is an orthonormal basis for the range of A. Q'*Q = I, the columns of Q span the same space as the columns of A and the number of columns of Q is the rank of A. See also QR, NULL.
Below is an example of using orth() to orthonormalize a linearly independent basis set for :
% Demonstration of the orth() function. v1 = [1; 2; 3]; % our first basis vector (a column vector) v2 = [1; -2; 3]; % a second, linearly independent vector v1' * v2 % show that v1 is not orthogonal to v2 ans = 6 V = [v1,v2] % Each column of V is one of our vectors V = 1 1 2 -2 3 3 W = orth(V) % Find an orthonormal basis for the same space W = 0.2673 0.1690 0.5345 -0.8452 0.8018 0.5071 w1 = W(:,1) % Break out the returned vectors w1 = 0.2673 0.5345 0.8018 w2 = W(:,2) w2 = 0.1690 -0.8452 0.5071 w1' * w2 % Check that w1 is orthogonal to w2 ans = 2.5723e-17 w1' * w1 % Also check that the new vectors are unit length ans = 1 w2' * w2 ans = 1 W' * W % faster way to do the above checks ans = 1 0 0 1 % Construct some vector x in the space spanned by v1 and v2: x = 2 * v1 - 3 * v2 x = -1 10 -3 % Show that x is also some linear combination of w1 and w2: c1 = x' * w1 % Coefficient of projection of x onto w1 c1 = 2.6726 c2 = x' * w2 % Coefficient of projection of x onto w2 c2 = -10.1419 xw = c1 * w1 + c2 * w2 % Can we make x using w1 and w2? xw = -1 10 -3 error = x - xw error = 1.0e-14 * 0.1332 0 0 norm(error) % typical way to summarize a vector error ans = 1.3323e-15 % It works! (to working precision, of course)
% Construct a vector x NOT in the space spanned by v1 and v2: y = [1; 0; 0]; % Almost anything we guess in 3D will work % Try to express y as a linear combination of w1 and w2: c1 = y' * w1; % Coefficient of projection of y onto w1 c2 = y' * w2; % Coefficient of projection of y onto w2 yw = c1 * w1 + c2 * w2 % Can we make y using w1 and w2?
yw = 0.1 0.0 0.3 yerror = y - yw yerror = 0.9 0.0 -0.3 norm(yerror) ans = 0.9487
While the error is not zero, it is the smallest possible error in the least squares sense. That is, yw is the optimal least-squares approximation to y in the space spanned by v1 and v2 (w1 and w2). In other words, norm(yerror) is less than or equal to norm(y-yw2) for any other vector yw2 made using a linear combination of v1 and v2. In yet other words, we obtain the optimal least squares approximation of y (which lives in 3D) in some subspace (a 2D subspace of 3D spanned by the columns of matrix W) by projecting y orthogonally onto the subspace to get yw as above.
An important property of the optimal least-squares approximation is that the approximation error is orthogonal to the the subspace in which the approximation lies. Let's verify this:
W' * yerror % must be zero to working precision ans = 1.0e-16 * -0.2574 -0.0119
This section gives Matlab examples illustrating the computation of two figures in Chapter 6, and the DFT matrix in Matlab.
DFT Sinusoids for
Below is the Matlab for Fig.6.2:
N=8; fs=1; n = [0:N-1]; % row t = [0:0.01:N]; % interpolated k=fliplr(n)' - N/2; fk = k*fs/N; wk = 2*pi*fk; clf; for i=1:N subplot(N,2,2*i-1); plot(t,cos(wk(i)*t)) axis([0,8,-1,1]); hold on; plot(n,cos(wk(i)*n),'*') if i==1 title('Real Part'); end; ylabel(sprintf('k=%d',k(i))); if i==N xlabel('Time (samples)'); end; subplot(N,2,2*i); plot(t,sin(wk(i)*t)) axis([0,8,-1,1]); hold on; plot(n,sin(wk(i)*n),'*') ylabel(sprintf('k=%d',k(i))); if i==1 title('Imaginary Part'); end; if i==N xlabel('Time (samples)'); end; end
DFT Bin Response
Below is the Matlab for Fig.6.3:
% Parameters (sampling rate = 1) N = 16; % DFT length k = N/4; % bin where DFT filter is centered wk = 2*pi*k/N; % normalized radian center-frequency wStep = 2*pi/N; w = [0:wStep:2*pi - wStep]; % DFT frequency grid interp = 10; N2 = interp*N; % Denser grid showing "arbitrary" frequencies w2Step = 2*pi/N2; w2 = [0:w2Step:2*pi - w2Step]; % Extra dense frequency grid X = (1 - exp(j*(w2-wk)*N)) ./ (1 - exp(j*(w2-wk))); X(1+k*interp) = N; % Fix divide-by-zero point (overwrite NaN) % Plot spectral magnitude clf; magX = abs(X); magXd = magX(1:interp:N2); % DFT frequencies only subplot(2,1,1); plot(w2,magX,'-'); hold on; grid; plot(w,magXd,'*'); % Show DFT sample points title('DFT Amplitude Response at k=N/4'); xlabel('Normalized Radian Frequency (radians per sample)'); ylabel('Magnitude (Linear)'); text(-1,20,'a)'); % Same thing on a dB scale magXdb = 20*log10(magX); % Spectral magnitude in dB % Since the zeros go to minus infinity, clip at -60 dB: magXdb = max(magXdb,-60*ones(1,N2)); magXddb = magXdb(1:interp:N2); % DFT frequencies only subplot(2,1,2); hold off; plot(w2,magXdb,'-'); hold on; plot(w,magXddb,'*'); xlabel('Normalized Radian Frequency (radians per sample)'); ylabel('Magnitude (dB)'); grid; text(-1,40,'b)'); print -deps '../eps/dftfilter.eps'; hold off;
The following example reinforces the discussion of the DFT matrix in §6.12. We can simply create the DFT matrix in matlab by taking the DFT of the identity matrix. Then we show that multiplying by the DFT matrix is equivalent to the calling the fft function in matlab:
>> eye(4) ans = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 >> S4 = fft(eye(4)) ans = 1 1 1 1 1 0 - 1i -1 0 + 1i 1 -1 1 -1 1 0 + 1i -1 0 - 1i >> S4' * S4 % Show that S4' = inverse DFT (times N=4) ans = 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4 >> x = [1; 2; 3; 4] x = 1 2 3 4
>> fft(x) ans = 10 -2 + 2i -2 -2 - 2i >> S4 * x ans = 10 -2 + 2i -2 -2 - 2i
This section lists the spectrogram function called in the Matlab code displayed in Fig.8.11.
function X = spectrogram(x,nfft,fs,window,noverlap,doplot,dbclip); %SPECTROGRAM Calculate spectrogram from signal. % B = SPECTROGRAM(A,NFFT,Fs,WINDOW,NOVERLAP) calculates the % spectrogram for the signal in vector A. % % NFFT is the FFT size used for each frame of A. It should be a % power of 2 for fastest computation of the spectrogram. % % Fs is the sampling frequency. Since all processing parameters are % in units of samples, Fs does not effect the spectrogram itself, % but it is used for axis scaling in the plot produced when % SPECTROGRAM is called with no output argument (see below). % % WINDOW is the length M window function applied, IN ZERO-PHASE % FORM, to each frame of A. M cannot exceed NFFT. For M<NFFT, % NFFT-M zeros are inserted in the FFT buffer (for interpolated % zero-phase processing). The window should be supplied in CAUSAL % FORM. % % NOVERLAP is the number of samples the sections of A overlap, if % nonnegative. If negative, -NOVERLAP is the "hop size", i.e., the % number of samples to advance successive windows. (The overlap is % the window length minus the hop size.) The hop size is called % NHOP below. NOVERLAP must be less than M. % % If doplot is nonzero, or if there is no output argument, the % spectrogram is displayed. % % When the spectrogram is displayed, it is "clipped" dbclip dB % below its maximum magnitude. The default clipping level is 100 % dB down. % % Thus, SPECTROGRAM splits the signal into overlapping segments of % length M, windows each segment with the length M WINDOW vector, in % zero-phase form, and forms the columns of B with their % zero-padded, length NFFT discrete Fourier transforms. % % With no output argument B, SPECTROGRAM plots the dB magnitude of % the spectrogram in the current figure, using % IMAGESC(T,F,20*log10(ABS(B))), AXIS XY, COLORMAP(JET) so the low % frequency content of the first portion of the signal is displayed % in the lower left corner of the axes. % % Each column of B contains an estimate of the short-term, % time-localized frequency content of the signal A. Time increases % linearly across the columns of B, from left to right. Frequency % increases linearly down the rows, starting at 0. % % If A is a length NX complex signal, B is returned as a complex % matrix with NFFT rows and % k = floor((NX-NOVERLAP)/(length(WINDOW)-NOVERLAP)) % = floor((NX-NOVERLAP)/NHOP) % columns. When A is real, only the NFFT/2+1 rows are needed when % NFFT even, and the first (NFFT+1)/2 rows are sufficient for % inversion when NFFT is odd. % % See also: Matlab and Octave's SPECGRAM and STFT functions. if nargin<7, dbclip=100; end if nargin<6, doplot=0; end if nargin<5, noverlap=256; end if nargin<4, window=hamming(512); end if nargin<3, fs=1; end if nargin<2, nfft=2048; end x = x(:); % make sure it's a column M = length(window); if length(x)<M, x = [x;zeros(M-length(x),1)]; end; if (M<2) % (Matlab's specgram allows window to be a scalar specifying % the length of a Hanning window.) error('spectrogram: Expect complete window, not just its length'); end; Modd = mod(M,2); % 0 if M even, 1 if odd Mo2 = (M-Modd)/2; w = window(:); % Make sure it's a column zp = zeros(nfft-M,1); wzp = [w(Mo2+1:M);zp;w(1:Mo2)]; noverlap = round(noverlap); % in case non-integer if noverlap<0 nhop = - noverlap; noverlap = M-nhop; else nhop = M-noverlap; end nx = length(x); nframes = 1+floor((nx-noverlap)/nhop); X = zeros(nfft,nframes); xoff = 0; for m=1:nframes-1 xframe = x(xoff+1:xoff+M); % extract frame of input data xoff = xoff + nhop; % advance in-pointer by hop size xzp = [xframe(Mo2+1:M);zp;xframe(1:Mo2)]; xw = wzp .* xzp; X(:,m) = fft(xw); end if (nargout==0) | doplot t = (0:nframes-1)*nhop/fs; f = 0.001*(0:nfft-1)*fs/nfft; Xdb = 20*log10(abs(X)); Xmax = max(max(Xdb)); % Clip lower limit so nulls don't dominate: clipvals = [Xmax-dbclip,Xmax]; imagesc(t,f,Xdb,clipvals); % grid; axis('xy'); colormap(jet); xlabel('Time (sec)'); ylabel('Freq (kHz)'); end
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