# Signal Filtering with Python

Notice

I have simplified and improved my ECG machine design! Check out the new post:

http://www.swharden.com/blog/2013-04-14-simple-diy-ecg-pulse-oximeter-version-2/

It’s time for a lecture. I’ve been spending a lot of time creating a DIY dlectrocardiogram and it produces fairly noisy signals. I’ve spent some time and effort researching the best ways to clean-up these signals, and the results are incredibly useful! Therefore, I’ve decided to lightly document these results in a blog entry.

Here’s an example of my magic! I take a noisy recording and turn it into a beautiful trace. See the example figure with the blue traces. How is this possible? Well I’ll explain it for you. Mostly, it boils down to eliminating excess high-frequency sine waves which are in the original recording due to electromagnetic noise. A major source of noise can be from the alternating current passing through wires traveling through the walls of your house or building. My original ECG circuit was highly susceptible to this kind of interference, but my improved ECG circuit eliminates most of this noise. However, noise is still in the trace (see the figure to the left), and it needed to be removed.

The key is the FFT (Fast Fourier Transformation) algorithm which can get pretty intimidating to research at first! I’ll simplify this process. Let’s say you have a trace with repeating sine-wave-shaped noise. The output of the FFT transformation of the signal is the breakdown of the signal by frequency. Check out this FFT trace of a noisy signal from a few posts ago (top graph). High peaks represent frequencies which are common. See the enormous peak around 60 Hz? (Hz means “per second” by the way, so 60 Hz is a sine wave that repeats 60 times a second) That’s from my AC noise. Other peaks (shown in colored bands) are other electromagnetic noise sources, such as wireless networks, TVs, telephones, and your computer processor. The heart produces changes in electricity that are very slow (the heartbeat is about 1 Hz, or 1 beat per second), so if we can eliminate all of the sine waves with frequencies higher than what we want to isolate we can get a pretty clear trace. This is called a band-stop filter (we block-out certain bands of frequencies) A band-pass filter is the opposite, where we only allow frequencies which are below (low-pass) or above (high-pass) a given frequency. By eliminating each of the peaks in the colored regions (setting each value to 0), then performing an inverse fast Fourier transformation (going backwards from frequency back to time), the result is the signal trace (seen as light gray on the bottom graph) with those high-frequency sine waves removed! (the gray trace on the bottom graph). A little touch-up smoothing makes a great trace (black trace on the bottom graph).

Here’s some Python code to get you started in cleaning-up your noisy signals! The image below is the output of the Python code at the bottom of this entry. This python file requires that test.wav (~700kb) (an actual ECG recording of my heartbeat) be saved in the same folder. Brief descriptions of each portion of the graph will follow.

(A) The original signal we want to isolate. (IE: our actual heart signal)

(B) Some electrical noise. (3 sine waves of different amplitudes and periods)

(C) Electrical noise (what happens when you add those 3 sine waves together)

(D) Static (random noise generated by a random number generator)

(E) Signal (A) plus static (D)

(F) Signal (A) plus static (D) plus electrical noise (C)

(G) Total FFT trace of (F). Note the low frequency peak due to the signal and electrical noise (near 0) and the high frequency peak due to static (near 10,000)

(H) This is a zoomed-in region of (F) showing 4 peaks (one for the original signal and 3 for high frequency noise). By blocking-out (set it to 0) everything above 10Hz (red), we isolate the peak we want (signal). This is a low-pass filter.

(I) Performing an inverse FFT (iFFT) on the low-pass iFFT, we get a nice trace which is our original signal!

(J) Comparison of our iFFT with our original signal shows that the amplitude is kinda messed up. If we normalize each of these (set minimum to 0, maximum to 1) they line up. Awesome!

(K) How close were we? Graphing the difference of iFFT and the original signal shows that usually we’re not far off. The ends are a problem though, but if our data analysis trims off these ends then our center looks great.

Here’s the code I used to make the image:

```
import numpy, scipy, pylab, random

# This script demonstrates how to use band-pass (low-pass)
# filtering to eliminate electrical noise and static
# from signal data!

##################
### PROCESSING ###
##################

xs=numpy.arange(1,100,.01) #generate Xs (0.00,0.01,0.02,0.03,...,100.0)
signal = sin1=numpy.sin(xs*.3) #(A)
sin1=numpy.sin(xs) # (B) sin1
sin2=numpy.sin(xs*2.33)*.333 # (B) sin2
sin3=numpy.sin(xs*2.77)*.777 # (B) sin3
noise=sin1+sin2+sin3 # (C)
static = (numpy.random.random_sample((len(xs)))-.5)*.2 # (D)
sigstat=static+signal # (E)
rawsignal=sigstat+noise # (F)
fft=scipy.fft(rawsignal) # (G) and (H)
bp=fft[:]
for i in range(len(bp)): # (H-red)
if i>=10:bp[i]=0
ibp=scipy.ifft(bp) # (I), (J), (K) and (L)

################
### GRAPHING ###
################

h,w=6,2
pylab.figure(figsize=(12,9))

pylab.subplot(h,w,1);pylab.title("(A) Original Signal")
pylab.plot(xs,signal)

pylab.subplot(h,w,3);pylab.title("(B) Electrical Noise Sources (3 Sine Waves)")
pylab.plot(xs,sin1,label="sin1")
pylab.plot(xs,sin2,label="sin2")
pylab.plot(xs,sin3,label="sin3")
pylab.legend()

pylab.subplot(h,w,5);pylab.title("(C) Electrical Noise (3 sine waves added together)")
pylab.plot(xs,noise)

pylab.subplot(h,w,7);pylab.title("(D) Static (random noise)")
pylab.plot(xs,static)
pylab.axis([None,None,-1,1])

pylab.subplot(h,w,9);pylab.title("(E) Signal + Static")
pylab.plot(xs,sigstat)

pylab.subplot(h,w,11);pylab.title("(F) Recording (Signal + Static + Electrical Noise)")
pylab.plot(xs,rawsignal)

pylab.subplot(h,w,2);pylab.title("(G) FFT of Recording")
fft=scipy.fft(rawsignal)
pylab.plot(abs(fft))
pylab.text(200,3000,"signals",verticalalignment='top')
pylab.text(9500,3000,"static",verticalalignment='top',
horizontalalignment='right')

pylab.subplot(h,w,4);pylab.title("(H) Low-Pass FFT")
pylab.plot(abs(fft))
pylab.text(17,3000,"sin1",verticalalignment='top',horizontalalignment='left')
pylab.text(37,2000,"sin2",verticalalignment='top',horizontalalignment='center')
pylab.text(45,3000,"sin3",verticalalignment='top',horizontalalignment='left')
pylab.text(6,3000,"signal",verticalalignment='top',horizontalalignment='left')
pylab.axvspan(10,10000,fc='r',alpha='.5')
pylab.axis([0,60,None,None])

pylab.subplot(h,w,6);pylab.title("(I) Inverse FFT")
pylab.plot(ibp)

pylab.subplot(h,w,8);pylab.title("(J) Signal vs. iFFT")
pylab.plot(signal,'k',label="signal",alpha=.5)
pylab.plot(ibp,'b',label="ifft",alpha=.5)

pylab.subplot(h,w,10);pylab.title("(K) Normalized Signal vs. iFFT")
pylab.plot(signal/max(signal),'k',label="signal",alpha=.5)
pylab.plot(ibp/max(ibp),'b',label="ifft",alpha=.5)

pylab.subplot(h,w,12);pylab.title("(L) Difference / Error")
pylab.plot(signal/max(signal)-ibp/max(ibp),'k')

pylab.savefig("SIG.png",dpi=200)
pylab.show()
```

Scott W Harden

Scott Harden has had a lifelong passion for computer programming and electrical engineering, and recently has become interested in its relationship with biomolecular sciences. He has run a personal website since he was 15, which has changed names from HardenTechnologies.com, to KnightHacker.com, to ScottIsHot.com, to its current SWHarden.com. Scott has been in college for 10 years, with 3 more years to go. He has an AA in Biology (Valencia College), BS in Cell Biology (Union University), MS in Molecular Biology and Microbiology (University of Central Florida), and is currently in a combined DMD (doctor of dental medicine) / PhD (neuroscience) program through the collaboration of the College of Dentistry and College of Medicine (Interdisciplinary Program in Biomedical Science, IDP) at the University of Florida in Gainesville, Florida. In his spare time Scott builds small electrical devices (with an emphasis on radio frequency) and enjoys writing cross-platform open-source software.

1. ##### Anonymous

I hope you can help or teach me how to smoothing my signal. data[0] is the realtime data and data[1] and data [2] is calculated from data [0]. because the initial data (data[0]) have a noise so that it disturb the other datas. Here is part of my program,

def update(): # Called periodically by the Tk toolkit
global data, running, timegap, ph
#s = time.time()
if running == False:
return
res = ph.get_voltage()
x = res[0] – starts[0]
v = res[1]
m = v2d(v)
data[0].append((x,m))
if len(data[0]) > 1: # plot position & calculate instantaneous velocity
dispobjects[0].delete_lines()
dispobjects[0].line(data[0], ‘black’)
vel = (data[0][-1][1] – data[0][-2][1]) / (data[0][-1][0] – data[0][-2][0])
data[1].append( (data[0][-1][0], vel) )

if len(data[1]) > 1: # plot velocity & calculate acceleration
dispobjects[1].delete_lines()
dispobjects[1].line(data[1], ‘black’)
acn = (data[1][-1][1] – data[1][-2][1]) / (data[1][0][-1] – data[1][0][-2])
data[2].append( (data[0][-1][0], acn) )

if len(data[2]) > 1: # plot acceleration
dispobjects[2].delete_lines()
dispobjects[2].line(data[2], ‘black’)
root.after(timegap, update)
if x > maxtime: running = False

nice to hear from yoyu soon

2. ##### Gopal Koduri

thanks babai (a kind of *dude* in my language ;))! I was looking for a similar thing(filtering) in python. This helped me get started :)

3. ##### CoreDistance

I think the wav file is not used in the code.

4. ##### Jphn

“This python file requires that test.wav (~700kb) (an actual ECG recording of my heartbeat) be saved in the same folder.”
=> false

Thanks for the code!

5. ##### Kamil

Thank you, great article!

6. ##### Danny

Thanks for the article, sure it’ll come in handy cleaning some of my (very noisy) data.

7. ##### Ken

Be careful with this method of “filtering.” This kind of direct manipulation of the spectrum results in time-aliasing when you take the inverse FFT. The result is not truly the original signal with its high frequencies filtered out, because the time-aliasing leaves the signal littered with artifacts.

The easiest way to do it properly would be to do a lowpass filter in the time-domain.

You can do a similar filter in the frequency domain (sort of like you’re doing), but the time-domain data and a filter impulse response needs to be properly zero-padded before you FFT them to avoid circular convolution (which results in time-aliasing).

8. ##### Anonymous

Thanks for the code snippet. I’ve got a few suggestions for you, though:
1) When I use the line “bp=fft[:]“, modifying bp still alters fft. I’ve used “bp = fft.copy()” instead.
2) When zeroing out FFT terms, you need to make the zeros symmetric around the mid-point of the FFT spectrum. The inverse FFT will have a non-negligible imaginary component if the frequency spectrum isn’t symmetric, even though the input data is strictly real.
3) To zero out fft coefficients, it’s easier to write bp[10:-10] = 0
4) To normalize the data, add the line bp *= real(fft.dot(fft))/real(bp.dot(bp)). fft.dot(fft) gives the total power in the signal, so this redistributes the power in the stop band more or less equally to the pass band.
5) It’s perhaps better to change the variable name ‘fft’ to something else so that it doesn’t conflict with numpy.fft.fft. If you’re working in iPython, numpy.fft.fft is automagically imported into the global namespace, so trying to use the fft function after running your script could cause a few headaches.

9. ##### Ahmet

Thanks for the information, but I was wondering something.
Maybe you can help me out with a problem. I have a signal of 0.1 seconds length which consists of 10.000 datapoints. First I averaged the datapoints 40 times such that it becomes a set of 250 datapoints. Even though this is a bit rough, the signal is still clearly visible.
When I make an FFT of this averaged signal the frequency resolution is too low; the frequency step is 10 Hz.
To increase the frequency resolution I decreased the averaging to 10 times (1000 datapoints), but I noticed that this does not matter for the frequency resolution; it stays 10 Hz.

So the frequency resolution is only determined by the measurement time (0.1s) and will not increase if I increase the number of datapoints during this 0.1 second…

My question is how I can increase the freq. resolution…

10. ##### Doug

Thank you for this excellent example of the power of numpy, scipy, and pylab!

1 more note. In graph (G) you label the right hand side peak as “static”. If you zoom in on the peak with:

pylab.axis([9840,9900,None,None])

You’ll see that it’s the symmetric part of your sine waves.

11. ##### Michael

I was thinking to make an ECG. I’m a real amateur. I don’t want to visualize my heartbeat but I want to transfer the signal of my heartbeat to a digital signal to use in a a microprocessor. I don’t wanne use a computer at all.
Can you pleas tell me how I could do it on a simple way? I wanne use the minimum of filters.

12. ##### Elliot

Thank you very much.
This was very helpful for me processing ultrasound signals.

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14. ##### Bogdan Hlevca

Great code, however, it has a serious weakness as it does not explain how the frequency value is selected for filtering. It can be inferred from the code, but there is a more elegant way of doing that, using the numpy functions instead of scipy functions.
Another problem that you have is the resulting value is half in amplitude. You need to multiply by 2 to get the expected value.

For example a filer function can be defined as:

``` import numpy as np```

``` def fft_bandpassfilter(data, fs, lowcut, highcut): fft = np.fft.fft(data) n = len(data) timestep = 1.0 / fs freq = np.fft.fftfreq(n, d = timestep) bp = fft[:] for i in range(len(bp)): if freq[i] >= highcut or freq[i] < lowcut: bp[i] = 0 ```

``` # must multipy by 2 to get the correct amplitude ibp = 2 * sp.ifft(bp) return ibp ```

where fs = sample frequency, data ids your timeseries and the other two the cutoff frequencies.

1. ##### Stu

Hi @Bogdan Hlevca
I’m trying to get fft_bandpassfilter working, is lowcut and highcut in HZ ?

+ frequencies within this range will be kept in ?

For sample frequency, I guess 44010 should work if I’m sampling pc microphone ?

Cheers
S

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17. ##### Niels

I would be nice if you warned readers about the flaws in this article. By removing the correct part of the FFT results, no scaling at all is needed. That’s the great thing about spectral filtering (al least in 1D, and preferably on something with cyclic boundaries): all the information beyond the cut-off frequency is maintained (plot some spectra of the data before and after filtering and you will see the result).

Anyhow, to warn readers: BE CAREFULL WITH THIS ARTICLE! NICE ATTEMPT, BUT WRITER IS BY FAR NO SPECIALIST, EXAMPLE CODE CONTAINS SERIOUS FLAWS!

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