Signals and Systems - Electrical Engineering

592 C H A P T E R 10: Fourier Analysis of Discrete-Time Signals and Systems

Phase unwrapping

Another problem with phase computation has to do with the way the phase is displayed as a function

of frequency with values within [−π,π)—thewrapped phase. If the DTFT of a signal is zero or infi-

nite at some frequencies, the phase at those frequencies is not determined, as any value would be

as good as any other because the magnitude is zero or infinity. On the other hand, if there are

no zeros or poles on the unit circle to make the magnitude zero or infinite at some frequencies,

the phase is continuous. However, because of the way the phase is computed and displayed with

values between−πtoπ, it seems discontinuous. These phase discontinuities are 2πwide, mak-

ing the phase values right before and right after the discontinuity identical. Finding the frequencies

where these discontinuities occur and patching the phase, it is possible to obtain the continuous

phase. The process is called theunwrapping of the phase. The MATLAB functionunwrapis used for this

purpose.

nExample 10.10

Consider a sinusoidx[n]=sin(πn/ 2 )to which we add a Gaussian noiseη[n] generated by the

MATLAB functionrandn, which theoretically can take any real value. Use the significance of the

magnitude computed by MATLAB to estimate the phase.

Solution

The DTFT ofx[n] consists of two impulses, one atπ/2 and the other at−π/2. Thus, the phase

should be zero everywhere except at these frequencies. However, any amount of added noise (even

in noiseless cases, given the computational precision of MATLAB) would give nonzero phases in

places where it should be zero.

In this case, since we know that the magnitude spectrum is significant at the negative and the

positive values of the sinusoid frequency, we use the magnitude as a mask to indicate where

the phase computation can be considered important given the significance of the magnitude.

The following script is used to illustrate the masking using the significance of the magnitude.

%%%%%%%%%%%%%%%%%%%%%%%%%

% Example 10.10---phase of sinusoid in noise

%%%%%%%%%%%%%%%%%%%%%%%%%

n = 0:99; x = sin(pi∗n/2) + 0.1∗randn(1,100); % sine plus noise

X = fftshift(fft(x)); % fft of signal

X1 = abs(X); theta = angle(X); % magnitude and phase

theta1 = theta.∗X1/max(X1); % masked phase

L = length(X); w = 0:2∗pi/L:2∗pi - 2∗pi/L;w1 = (w - pi)/pi; % frequency range

Using the mask, the noisy phase (Figure 10.6(b)) is converted into the phase of the sine, which

occurs when the magnitude is significant (Figures 10.5(a) and 10.5(c)). n