Qualitative nature of symmetry

Type I Positive symmetry, N is odd. To explain take N = 5: 

We have to included up a(0), and the two cosine terms. It is obvious that at ω = 0 all the cosine terms are at their positive side, so that when included the response of pass filter. Suppose |H (e^jw )| vs. ω could show a low pass filter

%Frequency response of moving average filter h(n) = {0.2, 0.2, 0.2, 0.2, 0.2}

b5 = [0.2, 0.2, 0.2, 0.2, 0.2], a = [1]

w=-pi: pi/256: pi; Hw5=freqz(b5, a, w);

subplot(2, 1, 1), plot(w, abs(Hw5)); legend ('Magnitude'); title ('Type I, N is odd'); xlabel('Frequency \omega, rad/sample'), ylabel('Magnitude of H(\omega)'); grid subplot(2, 1, 2), plot(w, angle(Hw5)); legend ('Phase');

xlabel('Frequency \omega, rad/sample'), ylabel('Phase of H(\omega)'); grid

Type II Positive symmetry, N is even. Take N = 6:

At w =Π, related to half the sampling frequency (maximum possible frequency), all the cosine parts will be zero. Thus that kind of filter is unsuitable as a high-pass filter. It could be ok as a low pass filter. Suppose

%Frequency response of moving average filter h(n) = {1/6, 1/6, 1/6, 1/6, 1/6, 1/6}

b6 = [1/6, 1/6, 1/6, 1/6, 1/6, 1/6], a = [1]

w=-pi: pi/256: pi; Hw6=freqz(b6, a, w);

subplot(2, 1, 1), plot(w, abs(Hw6)); legend ('Magnitude');

title ('Type II, N is even');

xlabel('Frequency \omega, rad/sample'), ylabel('Magnitude of H(\omega)'); grid subplot(2, 1, 2), plot(w, angle(Hw6)); legend ('Phase');

xlabel('Frequency \omega, rad/sample'), ylabel('Phase of H(\omega)'); grid

Type III Negative symmetry, N is odd. That defines a 90⁰ (= π/2) phase shift. Due to the sine terms |H| is always zero at ω = 0 and at ω = π/2 (half the sampling frequency). Therefore the filter is unsuitable as a high pass filter or a low pass. To explain take N = 5 and h(n) = {0.2, 0.2, 0, -0.2, -0.2}

%Frequency response of Type III filter, h(n) = {0.2, 0.2, 0, -0.2, -0.2}

b5 = [0.2, 0.2, 0, -0.2, -0.2], a = [1]

w=-pi: pi/256: pi; Hw5=freqz(b5, a, w);

subplot(2, 1, 1), plot(w, abs(Hw5)); legend ('Magnitude');

title ('Type III, N is odd');

xlabel('Frequency \omega, rad/sample'), ylabel('Magnitude of H(\omega)'); grid subplot(2, 1, 2), plot(w, angle(Hw5)); legend ('Phase');

xlabel('Frequency \omega, rad/sample'), ylabel('Phase of H(\omega)'); grid

Type IV Negative symmetry, N is even. That produces a 90⁰ (= π/2) phase shift. Due to the sine terms |H| is always zero at ω = 0. Therefore the filter is unsuitable as a low pass filter. To explain take N = 6 and

%Frequency response of Type IV filter h(n) = {1/6, 1/6, 1/6, -1/6, -1/6, -1/6}

b6 = [1/6, 1/6, 1/6, -1/6, -1/6, -1/6], a = [1]

w=-pi: pi/256: pi; Hw6=freqz(b6, a, w);

subplot(2, 1, 1), plot(w, abs(Hw6)); legend ('Magnitude');

title ('Type IV, N is even');

xlabel('Frequency \omega, rad/sample'), ylabel('Magnitude of H(\omega)'); grid subplot(2, 1, 2), plot(w, angle(Hw6)); legend ('Phase');

xlabel('Frequency \omega, rad/sample'), ylabel('Phase of H(\omega)'); grid

Types III and IV are usually used to describe differentiators and Hilbert transformers due to the 90⁰ phase shift that each one can give.

The phase delay for Type I and II filters or group delay for all types of filters is expressible in parts of the number of coefficients of the filter and so may be corrected to provide a zero phase or group delay response.

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