In pulse wave signal, which is a reflection of the activity of the heart, there are many types of noises that can affect the quality and accuracy of the recorded response. Both physiological or non-physiological factors can be the reason for these types of noises. Some common types of noises include:
• Baseline wander: Baseline wander or baseline drift is a low-frequency variation of the signal that can create signal distortion, decrease signal quality or missing detections, etc. This type of noise arises from breathing, body movement or poor electrodes connection.
• Power line interference: Power line interference is a common type of noise in electrical devices that is created by the power supply to the device or the neighboring power lines or electrical devices. It introduces a 50-60 Hz sinusoidal waveform, based on the region, that significantly contaminates the recorded data.
This contamination can obscure and distort the recorded signal that makes it difficult to accurately analyze and interpret the physiological data.
Figure 2.11: Signal with power line interference
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Figure 2.12: Signal with baseline wander
In order to remove these types of noises, many different approaches were proposed such as the finite impulse response filter, infinite impulse response filter, and the wavelet-based filter.
2.4.1. Finite Impulse Response (FIR) filter
A FIR is a filter whose impulse response is of finite duration since it reaches zero in a finite amount of time. In other words, it is a non-recursive filter that has no feedback in its equation.
By not utilizing feedback, FIR filters are inherently stable. This stability ensures that the output of the filter doesn't exhibit any unbounded or oscillatory behavior. Additionally, the phase issue of FIR filter can easily by designed to be linear phase by making the coefficient sequence symmetric; linear phase, or phase change proportional to frequency, corresponds to equal delay at all frequencies.
For a causal discrete-time FIR filter of order N, the output is calculated by convolving its input with its impulse response. The operation is described by the following equation:
𝑦[𝑛] = 𝑏0𝑥[𝑛] + 𝑏1𝑥[𝑛 − 1] + ⋯ + 𝑏𝑁𝑥[𝑛 − 𝑁] = ∑ 𝑏𝑖∙ 𝑥[𝑛 − 𝑖]
𝑁
𝑖=0
(2.5) Where:
o 𝑥[𝑛] is the input signal, o 𝑦[𝑛] is the output signal o 𝑁 is the filter order
o 𝑏𝑖 is the value of the impulse response at the ith instant for 0 ≤ 𝑖 ≤ 𝑁 of an 𝑁𝑡ℎ-order FIR filter
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Figure 2.13: A direct form discrete-time FIR filter of order N (Source: Wikipedia) 2.4.2. Infinite Impulse Response (IIR) filter
A IIR filter is a recursive filter whose output is computed by using the current and previous inputs and previous outputs. This mechanism is called a feedback loop. By utilizing it, the IIR filter can achieve more efficient frequency response shaping. The transfer function of an IIR filter, which is a polynomial ratio in the complex variable "z" determines the frequency response of the filter. The poles and zeros of the transfer function are the locations on the z-plane where the numerator and denominator polynomials, respectively, equal zero. The locations of poles and zeros in the z-plane significantly affect the characteristics of the IIR filter. While the former determines the frequency response shape, the latter can cancel or emphasize certain frequencies.
Based on this, IIR filters can be modified to achieve various frequency response characteristics such as low-pass, high-pass, band-pass, band-stop, by carefully choosing the pole and zero.
Mathematically, IIR filter is described as:
𝑦[𝑛] = 1
𝑎0( 𝑏0𝑥[𝑛] + 𝑏1𝑥[𝑛 − 1] + ⋯ + 𝑏𝑃𝑥[𝑛 − 𝑃]
−𝑎1𝑦[𝑛 − 1] − 𝑎2𝑦[𝑛 − 2] − ⋯ − 𝑎𝑄𝑦[𝑛 − 𝑄]) (2.6) Where:
o P is the feedforward filter order
o 𝑏𝑖 are the feedforward filter coefficients o 𝑄 is the feedback filter order
o 𝑎𝑖 are the feedback filter coefficients o 𝑥[𝑛] is the input signal
o 𝑦[𝑛] is the output signal
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2.4.3. Wavelet-based filter
Wavelet is a wave-like oscillation that is localized in time. It has two main properties which are scale and location. While the former defines the dilations, the latter defines the translations of a fixed function called the mother wavelet. In particular, a mother wavelet is a function satisfying two conditions which are zero mean and unit energy
∫ 𝜓(𝑥)𝑑𝑥 = 0
+∞
−∞
(2.7)
∫ 𝜓(𝑥)2𝑑𝑥 = 1
∞
−∞
(2.8) Based on the mother wavelet, the wavelet basis functions are derived as:
𝜓𝑎,𝑏(𝑡) = 1
√𝑏𝜓(𝑡 − 𝑎
𝑏 ) (2.9)
Where:
o a is a translation constant o b is dilation constant
By changing translation and dilation constant, wavelet transform becomes an ideal tool that is used to analyze signals in different frequencies with varying resolutions. This process is called multiresolution analysis.
The most fundamental and often used wavelet transform is the DWT, which is implemented by a bank of two-channel filters with varying levels. DWT is also known as dyadic wavelet transform since it is created by discretizing the scale and displacement of continuous wavelet transform according to powers of two. DWT decomposes the original signal into two parts including approximate coefficients, detail coefficients. While approximate coefficients act as a low-pass filter, detail coefficients represent the high-frequency information. DWT can be repeatedly operated on the approximate coefficients to achieve lower resolution components [5].
After each level of decomposition, the frequency bands of the signal are divided by two compared to the sample rate of the original signal. Additionally, the signal is also down-sampled by a factor of 2 after each level of decomposition.
Based on the problem, different types of mother wavelet can be used. Some wavelet families that are common in biomedical signal such as EEG, ECG, or pulse wave include daubechies, symlet, coiflet, morlet wavelet [3, 15, 16].
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Figure 2.14: Wavelet Families of discrete wavelets and continuous wavelets (Source: [4])
Normally, researchers have to conduct a comparison based on several metrics such as signal-to-noise ratio or mean squared error to choose the best wavelet families for the problem.