# How to read Phase response graph?

I'm a noob on DSP in general. But I'd like to understand this graph. It is the phase response of a band-stop filter, from Wikipedia:

I get the magnitude graph above: at some frequencies, the gain go down, than up, after the cut. But what about the phase?

I don't understand...

Why at the cutoff point, the frequencies (at that point) are drifted -90°/90° moved? What does it means? Is it the same phase as sin(90°) = 1?

I really don't understand that graph (visually). Is there any tool online maybe to show what is happening?

• Yes, you are absolutely right. That how it's usually calculated and plotted. You can unwrap it though. – jojek Oct 20 '16 at 18:00
• In this case, unwrapping wouldn't change much, because the phase is already in the interval $[-,\pi,\pi]$. – Matt L. Oct 20 '16 at 21:30

The frequency response of a filter (or, in general: of a linear time-invariant system) is a complex-valued function

$$H(j\omega)=M(\omega)e^{j\phi(\omega)}\tag{1}$$

where $M(\omega)$ is the magnitude response, and $\phi(\omega)$ is the phase response. The most straightforward way to interpret those two functions is to look at the way such a system responds to a sinusoidal input signal. For an input $x(t)=A\cos(\omega_0t)$, the response is given by

$$y(t)=AM(\omega_0)\cos(\omega_0t+\phi(\omega_0))\tag{1}$$

So, as you've correctly pointed out, the magnitude defines the filter's frequency dependent gain. The phase response determines the phase shift experienced by a sinusoidal input signal.

Assuming $\omega_0\neq 0$, Eq. $(1)$ can be rewritten as

$$y(t)=AM(\omega_0)\cos\left(\omega_0\left(t+\frac{\phi(\omega_0)}{\omega_0}\right)\right)=AM(\omega_0)\cos\left(\omega_0\left(t-\tau_p(\omega_0)\right)\right)\tag{2}$$

with the phase delay $\tau_p(\omega)$ defined by

$$\tau_p(\omega)=-\frac{\phi(\omega)}{\omega},\quad\omega\neq 0\tag{3}$$

From $(2)$ we see that a sinusoidal signal of frequency $\omega_0$ experiences a delay of $\tau_p(\omega_0)$. Note that the phase delay $\tau_p(\omega)$ is different from the group delay (cf. this wikipedia article).

A good visualization would be to look at formula and then a graph of the sum-of-sines trig identity.

The result of adding two sinewaves that are almost the same frequency results in a sinewave of about the same frequency, but modulated by a slow cosine wave of the frequency difference. The result of a variable input frequency sinewave and a fixed frequency sinewave (say, the frequency of the notch filter) that are close to the same frequency results in a sinewave within the same frequency range as the two, but modulated by the cosine of the difference. This cosine will have opposite signs depending on whether the input frequency is higher or lower than the fixed frequency (note the subtraction in the sum-of-sines formula), (the sum frequency will also change from higher to lower) and thus the phase of the sum-of-sines will be opposite as the variable frequency changes to opposite sides of the fixed frequency, due to the sign of the cosine modulator.

Why are we adding sinusoids? A notch filter is roughly equivalent to subtracting (and thus adding the inverse of) a sinewave at the frequency of the notch (and of a "suitable" amplitude) to the input in order to cancel out the spectral component at the notch frequency.

All other users said is indisputably correct. I wanna be little more conversational with your question. I think that a"nice way" to answer to this question is: «The Phase Response and the Magnitude Response are like Chili and Carne: you can relish both apart, but it will never taste like you eat together." I will show you better. The Magnitude Diagram tells you "how the magnitude of the output signal's frequencies" changes. The Phase Diagram, instead, tells how the single frequential components are late on the output. You can have many many types of phase responses but what you always have to do is: 1. When its progress is quite "linear", you have to look the "slope". Whereas the slope is more pronounced (high/low Degrees) the output frequencies are more (forward or backward) delayed than the others. 2. When its progress is not linear (as the case shown above) the filter works wrong around the frequency where it has "infinite delay" and where also the magnitude response has a "notch" behavoiur (corresponding to a frequency deletion).
Happy DSP to everyone!