Calculate phase difference between two general signals (e.g. HbO and Hb) using Hilbert transform

1 min read

In a recent fNIRS journal club (vedio recorded here), Dr. Tong talked about their work on the phase difference between oxy and deoxy Hb, and its relationship with participants’ age. This article is a demo of how to use Hilbert transform to calculate the phase difference between two signals, and whether it’s valid if the signals contain a wide range of frequencies.

Tool: We use MatLab’s hilbert function to calculate the instant phase of a signal. The code can be found in the end of this article.

Case 1: single frequency

Here we tried the simplest case: x and y are single frequency signals with phase difference pi/2, as demonstrated in the figure below. We expect that the calculated phase difference is close to pi/2 or 1.57.

As expected, the phase difference is pi/2! Hilbert transform is doing well in picking up the phase difference! It is noted, however, the calculation is more robust in the middle of the signal.

Case 2: 2 frequencies

What about more complex signals? If x and y both contains two frequencies, each with phase difference pi/2, will Hilbert transform find the correct value?

Yes. It turns out Hilbert transform works for two-frequency signals as well.

Case 3: a lot of frequencies

In this case we generated x and y each with 50 random frequencies. The resulted signals are essentially random:

Again, Hilbert transform found the correct phase difference, pi/2:

People may argue that in such a wide-band case the “phase difference” is not meaningful. However, if the phase difference for each frequency is similar, then it is reasonable to say that the phase difference between the overall signals exists and is that value.

Code (MatLab)

Update: in the original version I shift phase manually (and contains an error!). Now we use unwrap and wrapToPi, suggested by Yafeng Pan, which is easier to use and produce the correct phase shift. Thank Yafeng!

% Demo of Hilbert transform to calculate phase difference between two
% signals
% Xu Cui 2021/6/7

% single frequency
t = [0:0.01:100];
x = sin(t);
y = sin(t+pi/2);
figure; plot(t, x, t, y);
legend('x','y');

xh = hilbert(x);
yh = hilbert(y);

xphase = unwrap(angle(xh));
yphase = unwrap(angle(yh));

figure; plot(t, xphase, t, yphase,'.');

phase_diff = wrapToPi(xphase-yphase);
figure; plot(t, phase_diff, '.');

% 2 frequencies
t = [0:0.01:100];
x = sin(t) + sin(2*t);
y = sin(t+pi/2) + sin(2*t+pi/2);
figure; plot(t, x, t, y);
legend('x','y');

xh = hilbert(x);
yh = hilbert(y);

xphase = unwrap(angle(xh));
yphase = unwrap(angle(yh));

figure; plot(t, xphase, t, yphase,'.');

phase_diff = wrapToPi(xphase-yphase);
figure; plot(t, phase_diff, '.');

% a lot of frequencies
t = [0:0.01:100];
x = zeros(size(t));
y = zeros(size(t));
frequency = rand(1,50)*10;
for ii=1:length(frequency)
    f = frequency(ii);
    x = x + sin(f*t);
    y = y + sin(f*t+pi/2);
end

figure; plot(t, x, t, y);
legend('x','y');

xh = hilbert(x);
yh = hilbert(y);

xphase = unwrap(angle(xh));
yphase = unwrap(angle(yh));

figure; plot(t, xphase, t, yphase,'.');

phase_diff = wrapToPi(xphase-yphase);
figure; plot(t, phase_diff, '.');

第二十四期 fNIRS Journal Club 通知 2021/10/23,10:00am

来自韩国釜山国立大学Keum-Shik Hong教授团队的杨大林同学将为大家介绍如何结合深度学习和近红外扫描技术实现早期老年痴呆症的识别,并分享深度学习技术在相关方面的应用经验。热烈欢迎大家参与讨论。 时间: 北京时间2021年10月23日上午10:00地点: https://zoom.com房间号: 876 7722 5723 密码: 600106 杨大林要讲解的文章如下: Yang, D., Huang, R., Yoo, S.-H., Shin, M.-J., Yoon, J.-A., Shin, Y.-Il., Hong,...
Xu Cui
10 sec read

第二十三期 fNIRS Journal Club 视频

北京时间2021年9月25日10:00,上海师范大学的张明明博士为大家介绍团体 (三人或者之上) 超扫描领域的进展,以及他们最近发表的超扫描决策行为的文献。 https://www.storkapp.me/pubpaper/34335212 Youtube: https://youtu.be/aWkdYn1q3Jk Youku 优酷:https://v.youku.com/v_show/id_XNTgwOTEwOTE1Mg==.html 相关资源: 文献鸟(追踪科学文献)文献大分析(两分钟了解一个领域)
Xu Cui
6 sec read

第二十三期 fNIRS Journal Club 通知 2021/09/25,10:00am

上海师范大学的张明明博士将为大家介绍团体 (三人或者之上) 超扫描领域的进展,以及他们最近发表的团体超扫描决策行为的文献。热烈欢迎大家参与讨论。 时间: 北京时间2021年9月25日上午10:00地点: https://zoom.com房间号: 894 6339 2399 密码: 264149 张博士要讲解的文章如下: Zhang, Jia, Wang (2021) Interbrain Synchrony of Team Collaborative Decision-Making: An fNIRS...
Xu Cui
9 sec read

Leave a Reply

Your email address will not be published. Required fields are marked *