Phase angle of a complex number In calculating the phase angle of a complex number, z = Re(z) + j Im(z), a hand held calculator, uses the formula tan^-1 {Im( z) / Re( z)}, and returns an answer in range -Π/2 < angle ≤ Π/2. Hence for both (1-j) and (-1+j) the phase angle so calculated is Π/4. However, (1-j) is in the 4th quadrant with ∠(1 - j) = -Π/4 whereas (-1+j) is in the 2nd quadrant with ∠(-1 + j) = 3Π/4. MATLAB has function angle which takes into account the real and imaginary parts separately (and not their ratio) and calculates the "4-quadrant inverse tangent".

Example 1.6.2 To illustrate magnitude and phase plots of DTFT we take a = 0.8 in the exponential sequence x(n) = aⁿ u(n), |a| < 1, treated above. Therefore x(n) = (0.8)n  u(n). We only need plot over the interval -π ≤ ω ≤ π or 0 ≤ ω ≤ 2π. To show, visually,

periodicity we have plotted over -3π ≤ ω ≤ 3π. We have

X (e^jw ) = ∑aⁿe^-jwn  =  ∑(ae^- jw )ⁿ  = 1/1 - a e^- jw   = 1/1 - 0.8 e^{- jw}

In MATLAB program segment that follows the algebraic expression for

X (e ^jw ) =1/ (1-0.8*exp(-j*w)) as (1)// (1-0.8*exp(-j*w)) . Note that we have used „w? for ω and the plot ranges from -2Π to 2Π. Both ω and the phase, ÐX (w ) , are in radians. The parameter 'k' means that

the plot/display is in black "color".

subplot(2,1,1);fplot('abs((1)/(1-0.8*exp(-j*w)))', [-3*pi,3*pi], 'k'); xlabel('\omega');ylabel('Magnitude'); subplot(2,1,2);fplot('angle((1)/(1-0.8*exp(-j*w)))', [-3*pi,3*pi], 'k'); xlabel('\omega');ylabel('Phase');

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