Meaning of Phase response of a filter? In simple words?
Here is the same experiment done in my blog: https://poweidsplearningpath.blogspot.com/2020/04/chapter-51-meaning-of-general-linear.html



First, there is a tiny mistake in the question. Not all FIR filters have the properties of Linear Phase. Only the four types of FIR possess the properties. 1 By contrast, all of the IIR filters are not linear phases. 

For me, one better description of general linear phase (GLP) is constant group delay. By definition, group delay is negative derivative of phase 2 (In fact, the detail phase/delay relationship can be derived but I suggest we just accept them.). Then, Derivative (group delay) of a linear (phase) is constant and vise versa. 

Then, let's move on to the meaning of group delay.
group delay of a frequency represents the delay unit of the filter to that frequency.
So, the filter may treat different frequencies with different delay unit. For an extreme bad example of an non linear filter, the input signal 'do re mi' may becomes 're mi do' in the output. A GLP filter can guarantee that such wield condition will never happen.



Here I wrote an example. The example comes from the Chapter 5.1.2 in the bible of DSP3 and I just implemented the sample.

First, a IIR Filter with phase response like this is given.


Fig. 1.

Here is the group delay (negative derivative of phase response) and the magnitude response. Please note that I delay the frequency in +-0.2 pi for about 150 units. BTW, the filter is a low pass filter so signal higher than 0.8pi is expected to be filtered.



Fig. 2.

Then, let's input a test signal like 'do re mi'. The signal x[n] are '0.8pi, 0.2pi, 0.4pi' in order. The corresponding frequency response is also provided.



Fig. 3.

And here is the output signal. The signal becomes 'empty, 0.4pi, 0.2pi'. The signal component with 0.8pi is filtered out as expected. 



Fig. 4.

To make thing more clear, here I points the number of Fig. 3 and Fig. 4 together. For 0.2pi component, the group delay is about 6.39 units but the group delay of 0.2pi component is about 153 unit. The output signal can confirm the prediction from group delay response. That is why the 0.2pi component becomes that last in the output.



Fig. 5.

In summary, 


linear phase equals to constant group delay.
GLP FIR filter can guarantee such scenario will never happen. But IIR can never achieve GLP. (But with the same mag frequency spectrm requirement, IIR usually can achieves the spec with lower delay (but not constant) in comparison with FIR.)




Reference:


FIR filter with linear phase, 4 types
https://en.wikipedia.org/wiki/Group_delay_and_phase_delay
A. Oppenheim and R. Schafer, Discrete Time Signal Processing 3rd. 2009




Matlab code

%% System
% H1[z]
b1 = conv([1 -.98*exp(j*.8*pi)],[1 -.98*exp(-j*.8*pi)]);
a1 = conv([1 -.8*exp(j*.4*pi)],[1 -.8*exp(-j*.4*pi)]);
H1 = tf(b1,a1,-1,'Variable','z^-1');

% H2[z]
H2 = tf(1,1,-1,'Variable','z^-1');
for k = 1:4
    ck = 0.95*exp(j*(0.15*pi+0.02*pi*k));
    ck_conj = conj(ck);
    b_tmp = conv([ck_conj -1],[ck -1]);
    b_tmp = conv(b_tmp,b_tmp);
    a_tmp = conv([1 -1*ck],[1 -1*ck_conj]);
    a_tmp = conv(a_tmp,a_tmp);
    H_tmp = tf(b_tmp,a_tmp,-1,'Variable','z^-1');

    H2 = series(H2,H_tmp);
end

% H[z]
H = series(H1,H2);

% Zero-Pole Plot, Fig. 5.2
[b_h,a_h] = tfdata(H );
b_h = cell2mat(b_h);
a_h = cell2mat(a_h);

figure;
zplane(b_h,a_h);
suptitle('Zero-Pole Plot, Fig 5.2');

% System Response.
L=1000;
dw=2*pi/L;
w = -pi:dw:pi-dw;
HH=freqz(b_h,a_h,w);
mag=abs(HH);
phase=angle(HH);

% Fig. 5.3
figure;
subplot(2,1,1);
plot(w,phase);
xticks([-pi -0.8*pi -0.6*pi -0.4*pi -0.2*pi 0 0.2*pi 0.4*pi 0.6*pi 0.8*pi pi]);
xticklabels({'-\pi','-0.8\pi','-0.6\pi','-0.4\pi','-0.2\pi','0','0.2\pi','0.4\pi','0.6\pi','0.8\pi','\pi'});
xlim([-pi pi]);
yticks([-4 -2 -0 2 4]);
ylabel('ARG[H(e^(^j^w^)]');
xlabel('w');
title('Phase response');

subplot(2,1,2);
plot(w,unwrap(phase));
xticks([-pi -0.8*pi -0.6*pi -0.4*pi -0.2*pi 0 0.2*pi 0.4*pi 0.6*pi 0.8*pi pi]);
xticklabels({'-\pi','-0.8\pi','-0.6\pi','-0.4\pi','-0.2\pi','0','0.2\pi','0.4\pi','0.6\pi','0.8\pi','\pi'});
xlim([-pi pi]);
ylabel('arg[H(e^(^j^w^)]');
xlabel('w');
title('Unwrap Phase response');
suptitle('ARG/arg Plot, Fig 5.3');

% Fig. 5.4
figure;
subplot(2,1,1);
plot(w(1:end-1),-1*diff(unwrap(phase))./diff(w));
xticks([-pi -0.8*pi -0.6*pi -0.4*pi -0.2*pi 0 0.2*pi 0.4*pi 0.6*pi 0.8*pi pi]);
xticklabels({'-\pi','-0.8\pi','-0.6\pi','-0.4\pi','-0.2\pi','0','0.2\pi','0.4\pi','0.6\pi','0.8\pi','\pi'});
xlim([-pi pi]);
ylabel('grd[H(e^(^j^w^)]');
title('Group Delay');

subplot(2,1,2);
plot(w,mag);
xticks([-pi -0.8*pi -0.6*pi -0.4*pi -0.2*pi 0 0.2*pi 0.4*pi 0.6*pi 0.8*pi pi]);
xticklabels({'-\pi','-0.8\pi','-0.6\pi','-0.4\pi','-0.2\pi','0','0.2\pi','0.4\pi','0.6\pi','0.8\pi','\pi'});
xlim([-pi pi]);
ylabel('|H(e^(^j^w^)|');
title('Magnitude response');
suptitle('GD/mag Plot, Fig 5.4');

%% Signal
M = 60;
n = 0:M;
w = 0.54-0.46*cos(2*pi*n/M);

N = 512;
x1 = zeros(1,N);
x2 = zeros(1,N);
x3 = zeros(1,N);
dw = 2*pi/N;
w_freq = -pi:dw:pi-dw;

for i = 0:M

    x1(i+M) = w(i+1)*cos(0.2*pi*i);
    x2(i+2*M-1) = w(i+1)*cos(0.4*pi*i-pi/2);
    x3(i+1) = w(i+1)*cos(0.8*pi*i+pi/5);
end
x = x1+x2+x3;
X = abs(fft(x));
X = fftshift(X);

% Fig. 5.5
figure;
subplot(2,1,1);
plot(x);
title('x[n]');
xlim([0,300]);
subplot(2,1,2);
plot(w_freq,X);
xticks([-pi -0.8*pi -0.6*pi -0.4*pi -0.2*pi 0 0.2*pi 0.4*pi 0.6*pi 0.8*pi pi]);
xticklabels({'-\pi','-0.8\pi','-0.6\pi','-0.4\pi','-0.2\pi','0','0.2\pi','0.4\pi','0.6\pi','0.8\pi','\pi'});
xlim([-pi pi]);
ylabel('|H(e^(^j^w^)|');
title('DTFT of X');
suptitle('Input time/Freq., Fig 5.5');


%% Output
y = filter(b_h,a_h,x);

% Fig. 5.6
figure;
plot(y);
xlim([0,300]);
xlabel('n');
title('output y[n], Fig 5.6');


%= Compre the Delay sample point.
figure;
subplot(2,1,1);
plot(x);
xlim([0,300]);
xlabel('n');
ylabel('x[n]');
title('input');
subplot(2,1,2);
plot(y);
xlim([0,300]);
xlabel('n');
ylabel('y[n]');
title('output');