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Design FIR filter to reject unwanted frequencies

Let’s see how to design a simple digital FIR filter to reject unwanted frequencies in an incoming signal. As a per-requisite, I urge you to read through this post: Introduction to digital filter design

Background on transfer function

The transfer function of a system provides the underlying support for ascertaining vital system response characteristics without solving the difference equations of the system. As mentioned earlier, the transfer function of a generic filter in Z-domain is given by ratio of polynomials in z

\[H(z) = \frac{ \displaystyle{\sum_{i=0}^{M} b_k z^{-1}}}{ 1 + \displaystyle{\sum_{i=1}^{N} a_k z^{-1}} } \quad\quad (1)\]

The values of z when H(z) =0 are called zeros of H(z). The values of z when H(z) = ∞ are called poles of H(z).

It is often easy to solve for zeros {z_i} and poles {p_j}, when the polynomials in the numerator and denominator are expressed as resolvable factors.

\[H(z) = \frac{ \displaystyle{\sum_{i=0}^{M} b_k z^{-1}}}{ 1 + \displaystyle{\sum_{i=1}^{N} a_k z^{-1}} } = \frac{N(z)}{D(z)} = \frac{b_M}{a_N} \frac{(z – z_1)(z – z_2)\cdots (z – z_M)}{(z – p_1)(z – p_2)\cdots (z – p_N)} \quad\quad (2) \]

The zeros {z_i} are obtained by finding the roots of the equation

\[N(z) = 0 \quad\quad (3)\]

The poles {p_j} are obtained by finding the roots of the equation

\[D(z) = 0 \quad\quad (4) \]

Pole-zero plots are suited for visualizing the relationship between the Z-domain and the frequency response characteristics. As mentioned before, the frequency response of the system H(e^jω) can be computed by evaluating the transfer function H(z) at specific values of z = e^jω. Because the frequency response is periodic with period 2π, it is sufficient to evaluate the frequency response for the range -π <= ω < π (that is one loop around the unit circle on the z-plane starting from z=-1 and ending at the same point).

FIR filter design

FIR filters contain only zeros and no poles in the pole-zero plot (in fact all the poles sit at the origin for a causal FIR). For an FIR filter, the location of zeros of H(z) on the unit-circle nullifies specific frequencies. So, to design an FIR filter to nullify specific frequency ω, we just have to place the zeros at corresponding locations on the unit circle (z=e^jω) where the gain of the filter needs to be 0. Let’s illustrate this using an example.

For this illustration, we would use this audio file as an input to the filtering system. As a first step in the filter design process, we should understand the nature of the input signal. Discrete-time Fourier transform (DTFT) is a tool for analyzing the frequency domain characteristics of the given signal.

The following function is used to compute the DTFT of the sequence read from the audio file.

import numpy as np
import matplotlib.pyplot as plt

from math import ceil,log,pi,cos
from scipy.fftpack import fft,fftfreq,fftshift

def compute_DTFT(x,M=0):
    """
    Compute DTFT of the given sequence x
    M is the desired length for computing DTFT (optional).
    Returns the DTFT X[k] and corresponding frequencies w (omega) arranged as -pi to pi
    """
    N = max(M,len(x))
    N = 2**(ceil(log(N)/log(2)))
    
    X = fftshift(fft(x,N))
    w = 2*pi*fftshift(fftfreq(N))    
    return (X,w)

Let’s read the audio file, load the samples as a signal sequence , and plot the sequence in time-domain/frequency domain (using DTFT).

from scipy.io.wavfile import read

samplerate, x = read('speechWithNoise.wav')
duration = len(x)/samplerate
time = np.arange(0,duration,1/samplerate)

fig1, (ax1,ax2) = plt.subplots(nrows=2,ncols=1)
ax1.plot(time,x)
ax1.set_xlabel('Time (s)')
ax1.set_ylabel('Amplitude')
ax1.set_title('speechWithNoise.wav - x[n]')

(X,w)= compute_DTFT(x)
ax2.plot(w,abs(X))
ax2.set_xlabel('Normalized frequency (radians/sample)')
ax2.set_ylabel('|X[k]|')
ax2.set_title('Magnitude vs. Frequency')

*Figure 1: Time-domain and frequency domain characteristics of the given audio sample*

*Figure 1: Time-domain and frequency domain characteristics of the given audio sample*

The magnitude vs. frequency plot simply shows huge spikes at θ=±1.32344 radians. The location of the spikes are captured by using the numpy.argmax↗ function.

maxIndex = np.argmax(X)
theta = w[maxIndex]
print(theta)

Since a sinusoid can be mathematically represented as

\[x[n] = cos (\theta n) = \frac{1}{2}\left( e^{j \theta n} + e^{-j \theta n }\right) \quad\quad (5)\]

The two spikes at θ=±1.32344 radians in the frequency domain, will manifest as a sinusoidal signal in the time domain.

Zooming in the area between θ= ±0.4 radians, the frequency domain plot reveals a hidden signal.

*Figure 2: Hidden signal revealed in frequency domain*

Now, our goal is to design an FIR filter that should reject the sinusoid at θ=±1.32344 radians, so that only the hidden signal gets filtered in.

Since the sinusoid that we want to reject is occurring at some θ radians in the frequency domain, for the FIR filter design, we place two zeros at

\[z_1 = e^{j \theta} \quad\quad z_2 = e^{-j \theta}\quad\quad (6)\]

Therefore, the transfer function of the filter system is given by

\[\begin{aligned} H_f(z) &= \left( 1 – z_1 z^{-1}\right)\left(1 – z_2 z^{-1} \right) \\ &= \left( 1 – e^{j \theta} z^{-1}\right)\left(1 -e^{-j \theta}z^{-1} \right) \\ & = 1 – 2\;cos\left(\theta\right)z^{-1} + z^{-2} \end{aligned}\]

This is a second order FIR filter. The coefficients of the filter are

\[b_0 = 1,\; b_1 = – 2 cos (\theta),\; b_2 = 1 \text{ and } a_0=1\]

For the given problem, to should reject the sinusoid at θ=±1.32344 radians, we set θ=1.32344 in the filter coefficients above.

Filter in action

Filter the input audio signal through the designed filter and plot the filtered output in time-domain and frequency domain. The lfilter function from scipy.signal package↗ is utilized for the filtering operation.

from scipy.signal import lfilter
y_signal = lfilter(b, a, x)
fig3, (ax3,ax4) = plt.subplots(nrows=1,ncols=2)
ax3.plot(time,y_signal,'g')
ax3.set(title='Noise removed speech - y[n]',xlabel='Time (s)',ylabel='Amplitude')

(Y,w)= compute_DTFT(y_signal)
ax4.plot(w,abs(Y)/max(abs(Y)),'r')
ax4.set(title='Frequency content of Y',xlabel='Normalized frequency (radians/sample)',ylabel='Magnitude - |Y[k]|')

The filter has effectively removed the sinusoidal noise, as evident from both time-domain and frequency domain plots.

*Figure 4: Extracted speech and its frequency content*

Save the filtered output signal as .wav file for audio playback

from scipy.io.wavfile import write
output_data = np.asarray(y_signal, dtype=np.int16)#convert y to int16 format
write("noise_removed_output.wav", samplerate, output_data)

Characteristics of the designed filter

Let’s compute the double sided frequency response of the designed FIR filter. The frequency response of the designed FIR digital filter is computed using freqz function from the scipy.signal package↗.

from scipy.signal import freqz
b = [1,-2*cos(theta),1] #filter coefficients
a = [1]
w, h = freqz(b,a,whole=True)#frequency response h[e^(jw)]
#whole = True returns output for whole range 0 to 2*pi
#To plot double sided response, use fftshift
w = w - 2*np.pi*(w>=np.pi) #convert to range -pi to pi
w = fftshift(w)
h = fftshift(h)

Plot the magnitude response, phase response, pole-zero plot and the impulse response of the designed filter.

#Plot Magnitude-frequency response
fig2, (ax) = plt.subplots(nrows=2,ncols=2)
ax[0,0].plot(w,abs(h),'b')
ax[0,0].set(title='Magnitude response',xlabel='Frequency [radians/sample]',ylabel='Magnitude [dB]')
ax[0,0].grid();ax[0,0].axis('tight');

#Plot phase response
angles = np.unwrap(np.angle(h))
ax[0,1].plot(w,angles,'r')
ax[0,1].set(title='Phase response',xlabel='Frequency [radians/sample]',ylabel='Angles [radians]')
ax[0,1].grid();ax[0,1].axis('tight');

#Transfer function to pole-zero representation
from scipy.signal import tf2zpk
z, p, k = tf2zpk(b, a)

#Plot pole-zeros on a z-plane
from  matplotlib import patches
patch = patches.Circle((0,0), radius=1, fill=False,
                    color='black', ls='dashed')
ax[1,0].add_patch(patch)
ax[1,0].plot(np.real(z), np.imag(z), 'xb',label='Zeros')
ax[1,0].plot(np.real(p), np.imag(p), 'or',label='Poles')
ax[1,0].legend(loc=2)
ax[1,0].set(title='Pole-Zero Plot',ylabel='Real',xlabel='Imaginary')
ax[1,0].grid()

#Impulse response
#create an impulse signal
imp = np.zeros(20)
imp[0] = 1

from scipy.signal import lfilter
y_imp = lfilter(b, a, imp) #drive the impulse through the filter
ax[1,1].stem(y_imp,linefmt='-.')
ax[1,1].set(title='Impulse response',xlabel='Sample index [n]',ylabel='Amplitude')
ax[1,1].axis('tight')

*Figure 3: Magnitude response, phase response, pole-zero plot and impulse response of the designed second order FIR filter*

Questions

Use the comment form below to answer the following questions

1) Is the filter that we designed a lowpass, highpass, bandpass or a bandstop filter ?
2) To reject a single sinusoidal signal, why two zeros are needed in the above filter design ?
3) What do you understand from the phase response plotted above ?

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Digital filter design – Introduction

Key focus: Develop basic understanding of digital filter design. Learn about fundamentals of FIR and IIR filters and the design choices.

Analog filters and digital filters are the two major classification of filters, depending on the type of signal signal they process. An analog filter, processes continuous-time signal analog signals. Whereas, digital filters process sampled, discrete-time signals.

Design of digital filters, involve the use of both frequency domain and time domain techniques.This is because, the filter specifications are often specified in frequency domain and the implementation is done in time-domain in the form of difference equations. Z-transform and discrete-time frequency transform (DTFT) are typical tools used for frequency domain analysis of filters.

Linear Time-Invariant Causal filter

The choice of filter and the design process depends on design specification, application and the performance issues associates with them. In this section, the focus is on the principles used for designing linear time-invariant causal filters.

A linear system obeys the principle of superposition. This means that an arbitrary signal can be represented as the weighted sums of shifted unit impulse functions. In a linear time-invariant filtering system, the filter coefficients do not change with time. Therefore, if the input signal is time-shifted, there will be a corresponding time-shift in the output signal. The term causal implies that the output of the system depends only on the present and past samples of input or output, and not on the future samples. Causality is required for real-time applications.

Solution for LTI causal filtering

The input-output relationship in a linear time-invariant causal filter system (shown above) is mathematically described using the following difference equation

\[ \begin{align} \sum_{k=0}^{N} a_k y[n-k] &= \sum_{i=0}^{M} b_i x[n-i] \\ \Rightarrow y[n] &= \sum_{i=0}^{M} b_i x[n-i] – \sum_{k=1}^{N} a_k y[n-k] \end{align} \quad \quad (1) \]

where, x[n] and y[n] are the input and output (sampled discrete-time) signals of the filtering system, a_k and b_i are the coefficients of the filter that is programmed to certain values for achieving the given filtering task. The values of M and N determine the number of such coefficients, in other words, the filter has M zeros and N poles.

Linear time-invariant systems are completely characterized by it response to an impulse signal δ[n]. Therefore, the analysis and solution for a given filtering task, is easily achieved by using the impulse response h[n], which is the response of the LTI system to an impulse signal δ[n] .

Therefore, the solution for the recursive equation in [1] is found by exciting the input with an impulse , that is x[n]= δ[n].

\[h[n] = \sum_{i=0}^{M} b_i \delta[n-i] – \sum_{k=1}^{N} a_k h[n-k] \quad \quad (2) \]

Z-transform and DTFT

Z-transform and discrete-time frequency transform (DTFT) are important tools for analyzing difference equations and frequency response of filters. Z-transform converts a discrete-time signal into a complex frequency domain representation.

The impulse response of a discrete-time causal system is analyzed using the unilateral or one-sided Z-transform. The unilateral Z-transform of a discrete-time signal x(n) is given by

where, n is an integer and z is a complex number with magnitude r and complex argument ω in radians.

\[ z = re^{j \omega} = r \cdot \left[ cos(\omega) + j\; sin (\omega) \right] \quad \quad (4)\]

If we let |z| = r = 1, then the equation for Z-transform transforms into discrete-time Fourier transform.

The Z-plane contains all values of z, whereas, the unit circle (defined by |z|=r=1) contains only z=e ^jω values. Therefore, we can state that Z-transform may exist anywhere on the Z-plane and DTFT exists only on the unit circle. One period of DTFT is equivalent to one loop around the unit circle on the Z-plane.

One of the important properties of Z-transform is with regards to time-shifting of discrete-time samples. A delay of K samples in the time domain is equivalent to multiplication by z^-k in the Z-transform domain.

\[x[n-K] \rightleftharpoons z^{-k} X(z) \quad \quad (7)\]

Re-writing equation (1) in Z-domain,

Therefore, the transfer function of the generic digital filter is given by

From equations (5) and (6), the frequency response of the system can be computed by evaluating the transfer function H(z) on the unit circle (i.e, |z| = r =1 → z = e^jω)

IIR filter

From implementation standpoint, there are two classes of digital filters: infinite impulse response (IIR) and finite impulse response (FIR) filters.

When a_k ≠ 0 as in equation (2), the filter structure is characterized by the presence of feedback elements. Due to the presence of feedback elements, the impulse response of the filter may not become zero beyond certain point in time, but continues indefinitely and hence the name infinite impulse response (IIR) filter.

\[IIR: \quad \quad h[n] = \sum_{i=0}^{M} b_i \delta[n-i] – \sum_{k=1}^{N} a_k h[n-k], \quad a_k \neq 0 \quad \quad (11) \]

Therefore, equation (9) and (10) are essentially the transfer function and the frequency response of an IIR filtering system.

There exist different methods for implementing the filter structure. Examples include, direct form I structure, direct form II structure, lattice structure, transposition, state space representation etc.., Each method has its own advantage and disadvantage. For example, some method may be robust to precision issues, coefficient sensitivity, lesser memory and computation requirement. The methods are chosen depending on the application.

Figure 2 shows the direct form I signal flow graph for the generic IIR filter described by equation (1). The boxes represented by Z^-1 are unit delays. This is because, the Z-transform of unit delay is Z^-1

Figure 2: Direct form I signal flow graph for an IIR filter

FIR filter

When a_k=0 for all ks, there is no feedback in the filter structure, no poles in the pole-zero plot (in fact all the poles sit at the origin for a causal FIR). The equation is no longer recursive. The impulse response of such filter dies out (becomes zero) beyond certain point in time and hence the name finite impulse response (FIR) filter.

Setting x[n] = δ[n] and a_k=0 for all k, the impulse response of an FIR filter is given by,

\[FIR: \quad \quad h[n] = \sum_{i=0}^{M} b_i \delta[n-i] \quad \quad (13) \]

Evaluating the z-transform of impulse input x[n] = δ[n] , in Z-domain following mapping holds

From equation (13) and (14), the transfer function of the FIR filter in Z-domain is given by

The frequency response is given by evaluating the transfer function on the unit circle |z|=1.

The direct form I signal flow graph for the FIR filter is shown in Figure 3

Figure 3: Direct form I signal flow graph for FIR filter

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References

[1] Z- transform – MIT open course ware ↗

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Tag: LTI Systems

Design FIR filter to reject unwanted frequencies

Background on transfer function

FIR filter design

Filter in action

Characteristics of the designed filter

Questions

Similar topics

Books by the author

Digital filter design – Introduction

Linear Time-Invariant Causal filter

Solution for LTI causal filtering

Z-transform and DTFT

IIR filter

FIR filter

References

Books by the author

Similar topics