1. 3.3.3 Derivative-based operators to remove low-frequency artifacts

2. Continuous-time signals:
Differentiation  HPF (highpass filtering)
L[d/dt] = s
H(f)
X(t)
dx(t)/dt f
|H(f)|
Integration  LPF (lowpass filtering)
dt = 1/s
H(f)
X(t)
x(t)dt f
|H(f)|

3. Discrete-time signals:
Difference  HPF
Z[x(n) - x(n-1)] = (1 - z^-1) X(z)
H(z)
x(n)
y(n) f
|H(f)|
y(n) = [x(n) – x(n-1)]/T, T = sampling frequency
Z[x(n) + x(n-1)] = (1 + z^-1) X(z)
Summation  LPF (lowpass filtering)
H(z)
X(n)
x(n) + x(n-1) f
|H(f)|

4. Matlab command: freqz

5. Matlab command: freqz In MatLab, Normalized with respect to fs/2

6. ECG signal with baseline drift (, and high-frequency noise)
What we want：to eliminate the baseline drift.
Baseline drift
In this section, three derivative-based operators will be introduced.
They are Filter 1, Filter 2, & Filter 3.

7. Filter 1: First-order difference operator
Difference  HPF
Z[x(n) - x(n-1)] = (1 - z^-1) X(z)
H(z)
x(n)
y(n)
y(n) = [x(n) – x(n-1)]/T, T = sampling frequency

8. How to generate pole-zero plot in Matlab
(1. Use “Matlab help”
(2. Search “pole-zero plot”
(3. Choose “zero-Pole Analysis”
(4. Use “fvtool”
Example:
Fvtool([1 -1],[1])

9. Filter 1: First-order difference operator
>>help freqz
[H,F] = FREQZ(B,A,N,Fs) and [H,F] = FREQZ(B,A,N,'whole',Fs) return frequency vector F (in Hz), where Fs is the sampling frequency (in Hz).
B = [1 -1]; A = [1];
freqz(B,A,200,1000)

10. Estimation of the frequency components of ECG

11. Filter 1: frequency response
Frequency: f(T) < f(P) < f(QRS) < f(noise)
Amplification: A(T) < A(P) < A(QRS) < A(noise)
QRS (17 Hz)
P (5.6Hz)
T (4.5 Hz)
High-frequency noise
Baseline drift
Any problem in phase response?
No, because of phase linearity
Figure 3.22

12. Filter 1: the output Figure 3.24 Why do P and T waves disappear?
What is the gain for them?

14. Filter 2: How to attenuate HF as well as LF noise
H3(z)
x(n)
y3(n)
[H3 w3] = freqz([1/2,0,-1/2])

15. Filter 2 --- frequency response
[H3 w3] = freqz([1/2,0,-1/2])

16. Filter 2  frequency response
High-frequency noise
Figure 3.23

17. Filter 2  the output
Figure 3.25

18. Filter 3  y(n+1) = x(n+1) - x(n) + 0.995 * y(n)

20. Filter 3  frequency response

22. Filter 3:
Figure 3.26

23. Filter 3  frequency response
Figure 3.27

24. Filter 3  the output
Figure 3.28