Tag: FIR filter

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 {zi} and poles {pj}, 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 {zi} are obtained by finding the roots of the equation

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

The poles {pj} 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=ejω) 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')
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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Similar topics

Essentials of Signal Processing
● Generating standard test signals
 □ Sinusoidal signals
 □ Square wave
 □ Rectangular pulse
 □ Gaussian pulse
 □ Chirp signal
● Interpreting FFT results - complex DFT, frequency bins and FFTShift
 □ Real and complex DFT
 □ Fast Fourier Transform (FFT)
 □ Interpreting the FFT results
 □ FFTShift
 □ IFFTShift
● Obtaining magnitude and phase information from FFT
 □ Discrete-time domain representation
 □ Representing the signal in frequency domain using FFT
 □ Reconstructing the time domain signal from the frequency domain samples
● Power spectral density
● Power and energy of a signal
 □ Energy of a signal
 □ Power of a signal
 □ Classification of signals
 □ Computation of power of a signal - simulation and verification
● Polynomials, convolution and Toeplitz matrices
 □ Polynomial functions
 □ Representing single variable polynomial functions
 □ Multiplication of polynomials and linear convolution
 □ Toeplitz matrix and convolution
● Methods to compute convolution
 □ Method 1: Brute-force method
 □ Method 2: Using Toeplitz matrix
 □ Method 3: Using FFT to compute convolution
 □ Miscellaneous methods
● Analytic signal and its applications
 □ Analytic signal and Fourier transform
 □ Extracting instantaneous amplitude, phase, frequency
 □ Phase demodulation using Hilbert transform
● Choosing a filter : FIR or IIR : understanding the design perspective
 □ Design specification
 □ General considerations in design

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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, ak and bi 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.

\[ z = re^{j \omega} = r \cdot \left[ cos(\omega) + j\; sin (\omega) \right] \quad \quad (4)\]
Figure 1: Z-transform and Z-plane

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 = ejω)

IIR filter

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

When ak ≠ 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 ak=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 ak=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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Similar topics

Essentials of Signal Processing
● Generating standard test signals
 □ Sinusoidal signals
 □ Square wave
 □ Rectangular pulse
 □ Gaussian pulse
 □ Chirp signal
● Interpreting FFT results - complex DFT, frequency bins and FFTShift
 □ Real and complex DFT
 □ Fast Fourier Transform (FFT)
 □ Interpreting the FFT results
 □ FFTShift
 □ IFFTShift
● Obtaining magnitude and phase information from FFT
 □ Discrete-time domain representation
 □ Representing the signal in frequency domain using FFT
 □ Reconstructing the time domain signal from the frequency domain samples
● Power spectral density
● Power and energy of a signal
 □ Energy of a signal
 □ Power of a signal
 □ Classification of signals
 □ Computation of power of a signal - simulation and verification
● Polynomials, convolution and Toeplitz matrices
 □ Polynomial functions
 □ Representing single variable polynomial functions
 □ Multiplication of polynomials and linear convolution
 □ Toeplitz matrix and convolution
● Methods to compute convolution
 □ Method 1: Brute-force method
 □ Method 2: Using Toeplitz matrix
 □ Method 3: Using FFT to compute convolution
 □ Miscellaneous methods
● Analytic signal and its applications
 □ Analytic signal and Fourier transform
 □ Extracting instantaneous amplitude, phase, frequency
 □ Phase demodulation using Hilbert transform
● Choosing a filter : FIR or IIR : understanding the design perspective
 □ Design specification
 □ General considerations in design

Precoding for partial response signaling schemes

Introduction to precoding

Intersymbol interference (ISI) is a common problem in telecommunication systems, such as terrestrial television broadcasting, digital data communication systems, and cellular mobile communication systems. Dispersive effects in high-speed data transmission and multipath fading are the main reasons for ISI. To maximize the capacity, the transmission bandwidth must be extended to the entire usable bandwidth of the channel and that also leads to ISI.

To mitigate the effect of ISI, equalization techniques can be applied at the receiver side. Under the assumption of correct decisions, a zero-forcing decision feedback equalization (ZF-DFE) completely removes the ISI and leaves the white noise uncolored. It was also shown that ZF-DFE in combination with powerful coding techniques, allows transmission to approach the channel capacity [1]. DFE is adaptive and works well in the presence of spectral nulls and hence suitable for various PR channels that has spectral nulls. However, DFE suffers from error propagation and is not flexible enough to incorporate itself with powerful channel coding techniques such as trellis-coded modulation (TCM) and low-density parity codes (LDPC).

This article is part of the book Wireless Communication Systems in Matlab, ISBN: 978-1720114352 available in ebook (PDF) format (click here) and Paperback (hardcopy) format (click here).

These problems can be practically mitigated by employing precoding techniques at the transmitter side. Precoding eliminates error propagation effects at the source if the channel state information is known precisely at the transmitter. Additionally, precoding at transmitter allows coding techniques to be incorporated in the same way as for channels without ISI. In this text, a partial response (PR) signaling system is taken as an example to demonstrate the concept of precoding.

Precoding system using filters

In a PR signaling scheme, a filter is used at the transmitter to introduce a controlled amount of ISI into the signal. The introduced ISI can be compensated for, at the receiver by employing an inverse filter . In the case of PR1 signaling, the filters would be

Generally, the filter is chosen to be of FIR type and therefore its inverse at the receiver will be of IIR type. If the received signal is affected by noise, the usage of IIR filter at the receiver is prone to error propagation. Therefore, instead of compensating for the ISI at the receiver, a precoder can be implemented at the transmitter as shown in Figure 1.

Figure 1: A pre-equalization system incorporating a modulo-M precoder

Since the precoder is of IIR type, the output can become unbounded. For example, let’s filter a binary data sequence through the precoder used for PR1 signaling scheme .

% Matlab code snippet
>> d=[1,0,1,0,1,0,1,0,1,0]
>> filter(1,[1 1],d)
ans = 1  -1  2  -2  3  -3  4  -4  5  -5

The result indicates that the output becomes unbounded and some additional measure has to be taken to limit the output. Assuming M-ary signaling schemes like MPAM is used for transmission, the unbounded output of the precoder can be bounded by incorporating modulo-M operation.

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Reference

[1] R. Price, Nonlinear Feedback Equalized PAM versus Capacity for Noisy Filter Channels, in Proceedings of the Int. Conference on Comm. (ICC ’72), 1972, pp. 22.12-22.17

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Topics in this chapter

Pulse Shaping, Matched Filtering and Partial Response Signaling
● Introduction
● Nyquist Criterion for zero ISI
● Discrete-time model for a system with pulse shaping and matched filtering
 □ Rectangular pulse shaping
 □ Sinc pulse shaping
 □ Raised-cosine pulse shaping
 □ Square-root raised-cosine pulse shaping
● Eye Diagram
● Implementing a Matched Filter system with SRRC filtering
 □ Plotting the eye diagram
 □ Performance simulation
● Partial Response Signaling Models
 □ Impulse response and frequency response of PR signaling schemes
● Precoding
 □ Implementing a modulo-M precoder
 □ Simulation and results

Choosing FIR or IIR ? Understand design perspective

“What is the best filter that I should use? FIR or IIR ?” is often the question asked by many. There exists two different types of Linear Time Invariant (LTI) filters from transfer function standpoint : FIR (Finite Impulse Response) and IIR (Infinite Impulse Response) filters and myriad design techniques for designing them.  The mere fact that there exists so many techniques for designing a filter, suggests that there is no single optimal filter design.  One has to weigh-in the pros and cons of choosing a filter design by considering the factors discussed here.

Before proceeding to know the secrets of choosing a filter, I urge you to brush up the fundamentals of digital filter design.

Design specification:

A filter design starts with a specification. We may have a simple specification that just calls for removing an unwanted frequency component or it can be a complete design specification that calls for various parameters like – amount of ripples allowed in passband, stop band attenuation, transition width etc.., The design specification usually calls for satisfying one or more of the following:

● Desired Magnitude response –
● Desired Phase response –
● Tolerance specifications – that specifies how much the filter response is allowed to vary when compared with ideal response. Examples include how much ripples allowed in passband, stop band etc..,

Figure 1: FIR and IIR filters can be realized as direct-form structures. (a) Structure with feed-forward elements only – typical for FIR designs. (b) Structure with feed-back paths – generally results in IIR filters. Other structures like lattice implementations are also available.

Given the specification above, the goal of a filter design process to choose the parameters , , and , such that the transfer function of the filter

yields the desired response: . In other words, the design process also involves choosing the number and location of the zeros zeros () and poles () in the pole-zero plot.

Two types of filter can manifest from the given transfer function above.

● When , there is no feedback in the filter structure, no poles in the pole-zero plot (in fact all the poles sit at the origin). The impulse response of such filter dies out (becomes zero) beyond certain point of time and it is classified as Finite Impulse Response (FIR) filter. It provides linear phase characteristic in the passband.
● When , 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, but continues indefinitely and hence the name Infinite Impulse Response (IIR) filter.
● Caution: In most cases, the presence of feedback elements provide infinite impulse response. It is not always true. There are some exceptional cases where the presence of feedback structure may result in finite impulse response. For example, a moving average filter will have a finite impulse response. The output of a moving average filter can be described using a recursive formula, which will result in a structure with feedback elements.

General considerations in design:

As specified earlier, the choice of filter and the design process depends on design specification, application and the performance issues associates with them. However, the following general considerations are applied in practical design.

Minimizing number of computations

In order to minimize memory requirements for storing the filter co-efficients , and to minimize the number of computations, ideally we would like to be as small as possible. For the same specification, IIR filters result in much lower order when compared to its FIR counter part. Therefore, IIR filters are efficient when viewed from this standpoint.

Need for real-time processing

The given application may require processing of input samples in real-time or the input samples may exist in a recorded state (example: video/audio playback, image processing applications, audio compression). From this perspective, we have two types of filter systems

● Causal Filter
— Filter output depends on present and past input samples, not on the future samples. The output may also depend on the past output samples, as in IIR filters. Strictly no future samples.
— Such filters are very much suited for real-time applications.

● Non-Causal Filter
— There are many practical cases where a non-causal filter is required. Typically, such application warrants some form of post-processing, where the entire data stream is already stored in memory.
— In such cases, a filter can be designed that can take in all type of input samples : present, past and future, for processing. These filters are classified as non-causal filters.
— Non-causal filters have much simpler design methods.

It can be often seen in many signal processing texts, that the causal filters are practically realizable. That does not mean non-causal filters are not practically implementable. In fact both types of filters are implementable and you can see them in many systems today. The question you must ask is : whether your application requires real-time processing or processing of pre-recorded samples. If the application requires real-time processing, causal filters must be used. Otherwise, non-causal filters can be used.

Consequences of causal filter:

If the application requires real-time processing, causal filters are the only choice for implementation. Following consequences must be considered if causality is desired.

Ideal filters with finite bands of zero response (example: brick-wall filters), cannot be implemented using causal filter structure. A direct consequence of causal filter is that the response cannot be ideal. Hence, we must design the filter that provides a close approximation to the desired response . If tolerance specification is given, it has to be met.

For example, in the case of designing a low pass filter with given passband frequency () and stopband frequencies (), additional tolerance specifications like allowable passband ripple factor (), stopband ripple factor () need to be considered for the design,. Therefore, the practical filter design involves choosing and and then designing the filter with and that satisfies all the given requirements/responses. Often, iterative procedures may be required to satisfy all the above (example: Parks and McClellan algorithm used for designing optimal causal FIR filters [1]).

Figure 2: A sample filter design specification

For a causal filter, frequency response’s real part and the imaginary part become Hilbert transform pair [2]. Hence the magnitude and phase responses become interdependent.

Stability

For a causal LTI digital filter will be BIBO (Bounded Input Bounded Output) stable, if and only if the impulse response is absolutely summable.

Impulse response of FIR filters are always bounded and hence they are inherently stable. On the other hand, an IIR filter may become unstable if not designed properly.

Consider an IIR filter implemented using a floating point processor that has enough accuracy to represent all the coefficients in the transfer function below

The corresponding impulse response is plotted in Figure 3(a). The plot shows that the impulse response decays rapidly to zero as increases. For this case, the sum in equation (2) will be finite. Hence this IIR filter is stable.

Suppose, if we were to implement the same filter in a fixed point processor and we are forced to round-off the co-efficients to 2 digits after the decimal point, the  same transfer function looks like this

The corresponding impulse response  plotted in Figure 3(b)  shows that the impulse response increases rapidly towards a constant value as  increases. For this case, the sum in equation (2) will tend to infinity. Hence the implemented IIR filter is unstable.

Figure 3: Impact of poorly implemented IIR filter on stability. (a) Stable IIR filter, (b) The same IIR filter becomes unstable due to rounding effects.

Therefore, it is imperative that an IIR filter implementation need to be tested for stability.  To analyze the stability of the filter, the infinite sum in equation (2) need to be computed and it is often difficult to compute this sum. Analysis of pole-zero plot is an alternate solution for this problem. To have a stable causal filter, the poles of the transfer function should lie completely strictly inside the unit circle on the pole-zero plot. The pole-zero plot for the above given transfer functions , are plotted in Figure 4. It shows that for the transfer function , all the poles lie within the unit circle (the region of stability) and hence it is a stable IIR filter. On the other hand,  for the transfer function ,  one poles lie exactly on the unit circle (ie, it is just out of the region of stability) and hence it is an unstable IIR filter.

Figure 4: Impact of poorly implemented IIR filter on stability. (a) Stable IIR filter since all poles lie inside the unit circle, (b) Due to rounding effects, the one pole lies on the unit circle (just outside the region of stability), hence it becomes unstable.

Linear phase requirement

In many signal processing applications, it is needed that a digital filter should not alter the angular relationship between the real and imaginary components of a signal, especially in the passband. In otherwords, the phase relationship between the signal components should be preserved in the filter’s passband.  If not, we have phase distortion.

Phase distortion is a concern in many signal processing applications. For example, in phase modulations like GMSK [3],the entire demodulation process hinges on the phase relationship between the inphase and quadrature components of the incoming signal. If we have a phase distorting filter in the demodulation chain, the entire detection process goes for a toss. Hence, we have to pay attention to the phase characteristics of such filters. To have no phase distortion when processing a signal through a filter, every spectral component of the signal inside the passband should be delayed by the same amount time delay measured in samples. In other words, the phase response in the passband should be a linear function (straight line) of frequency (except for the phase wraps at the band edges).  A filter that satisfies this property is called a linear phase filter. FIR filters provide perfect linear phase characteristic in the passband region (Figure 5) and hence avoids phase distortion. All IIR filters provide non-linear phase characteristic. If a real-time application warrants for zero phase distortion, FIR filters are the immediate choice for design.

It is intuitive to see the phase response of a generalized linear phase filter should follow the relationship , where is the slope and is the intercept when viewing the linear relationship between the frequency and the phase response in the passband (Figure 5). The phase delay and group delay are the important filter characteristics considered for ascertaining the phase distortion and they relate to the intercept and the slope of the phase response in the passband. Linear phase filters are characterized by constant group delay. Any deviation in the group delay from the constant value inside the passband, indicates presence of certain degree of non-linearity in the phase and hence causes phase distortion.

Figure 5: FIR filter showing linear phase characteristics in the passband

Phase delay is the time delay experienced by each spectral component of the input signal. For a filter with the frequency response , the phase delay response defined in terms of phase response as

Group delay is the delay experienced by a group of spectral components within a narrow frequency interval about [4]. The group delay response is defined as the negative derivative of the phase response .

For the generalized linear phase filter, the group delay and phase delay are given by

Summary of design choices

● IIR filters are efficient, they can provide similar magnitude response for fewer coefficients or lower sidelobes for same number of coefficients
● For linear phase requirement, FIR filters are the immediate choice for the design
● FIR filters are inherently stable. IIR filters are susceptible to finite length words effects of fixed point arithmetic and hence the design has to be rigorously tested for stability.
● IIR filters provide less average delay compared to its equivalent FIR counterpart. If the filter has to be used in a feedback path in a system, the amount of filter delay is very critical as it affects the stability of the overall system.
● Given a specification, an IIR design can be easily deduced based on closed-form expressions. However, satisfying the design requirements using an FIR design generally requires iterative procedures.

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References:

[1] J. H. McClellan, T. W. Parks, and L. R. Rabiner, “A Computer Program for Designing Optimum FIR Linear Phase Digital Filters,” IEEE Trans, on Audio and Electroacoustics, Vol. AU-21, No. 6, pp. 506-526, December 1973.↗

[2] Frank R. Kschischang, ‘The Hilbert Transform’, Department of Electrical and Computer Engineering, University of Toronto, October 22, 2006.↗

[3]  Thierry Turletti, ‘GMSK in a nutshell’, Telemedia Networks and Systems Group Laboratory for Computer Science, Massachussets Institute of Technology April, 96.↗

[4] Julius O. Smith III, ‘Introduction to digital filters – with audio applications’, Center for Computer Research in Music and Acoustics (CCRMA), Department of Music, Stanford University.↗

Topics in this chapter

Essentials of Signal Processing
● Generating standard test signals
 □ Sinusoidal signals
 □ Square wave
 □ Rectangular pulse
 □ Gaussian pulse
 □ Chirp signal
● Interpreting FFT results - complex DFT, frequency bins and FFTShift
 □ Real and complex DFT
 □ Fast Fourier Transform (FFT)
 □ Interpreting the FFT results
 □ FFTShift
 □ IFFTShift
● Obtaining magnitude and phase information from FFT
 □ Discrete-time domain representation
 □ Representing the signal in frequency domain using FFT
 □ Reconstructing the time domain signal from the frequency domain samples
● Power spectral density
● Power and energy of a signal
 □ Energy of a signal
 □ Power of a signal
 □ Classification of signals
 □ Computation of power of a signal - simulation and verification
● Polynomials, convolution and Toeplitz matrices
 □ Polynomial functions
 □ Representing single variable polynomial functions
 □ Multiplication of polynomials and linear convolution
 □ Toeplitz matrix and convolution
● Methods to compute convolution
 □ Method 1: Brute-force method
 □ Method 2: Using Toeplitz matrix
 □ Method 3: Using FFT to compute convolution
 □ Miscellaneous methods
● Analytic signal and its applications
 □ Analytic signal and Fourier transform
 □ Extracting instantaneous amplitude, phase, frequency
 □ Phase demodulation using Hilbert transform
● Choosing a filter : FIR or IIR : understanding the design perspective
 □ Design specification
 □ General considerations in design

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Fading channel – complex baseband equivalent models

Keyfocus: Fading channel models for simulation. Learn how fading channels can be modeled as FIR filters for simplified modulation & detection. Rayleigh/Rician fading.

Introduction

A fading channel is a wireless communication channel in which the quality of the signal fluctuates over time due to changes in the transmission environment. These changes can be caused by different factors such as distance, obstacles, and interference, resulting in attenuation and phase shifting. The signal fluctuations can cause errors or loss of information during transmission.

Fading channels are categorized into slow fading and fast fading depending on the rate of channel variation. Slow fading occurs over long periods, while fast fading happens rapidly over short periods, typically due to multipath interference.

To overcome the negative effects of fading, various techniques are used, including diversity techniques, equalization, and channel coding.

Fading channel in frequency domain

With respect to the frequency domain characteristics, the fading channels can be classified into frequency selective and frequency-flat fading.

A frequency flat fading channel is a wireless communication channel where the attenuation and phase shift of the signal are constant across the entire frequency band. This means that the signal experiences the same amount of fading at all frequencies, and there is no frequency-dependent distortion of the signal.

In contrast, a frequency selective fading channel is a wireless communication channel where the attenuation and phase shift of the signal vary with frequency. This means that the signal experiences different levels of fading at different frequencies, resulting in a frequency-dependent distortion of the signal.

Frequency selective fading can occur due to various factors such as multipath interference and the presence of objects that scatter or absorb certain frequencies more than others. To mitigate the effects of frequency selective fading, various techniques can be used, such as equalization and frequency hopping.

The channel fading can be modeled with different statistics like Rayleigh, Rician, Nakagami fading. The fading channel models, in this section, are utilized for obtaining the simulated performance of various modulations over Rayleigh flat fading and Rician flat fading channels. Modeling of frequency selective fading channel is discussed in this article.

Linear time invariant channel model and FIR filters

The most significant feature of a real world channel is that the channel does not immediately respond to the input. Physically, this indicates some sort of inertia built into the channel/medium, that takes some time to respond. As a consequence, it may introduce distortion effects like inter-symbol interference (ISI) at the channel output. Such effects are best studied with the linear time invariant (LTI) channel model, given in Figure 1.

Figure 1: Complex baseband equivalent LTI channel model

In this model, the channel response to any input depends only on the channel impulse response(CIR) function of the channel. The CIR is usually defined for finite length \(L\) as \(\mathbf{h}=[h_0,h_1,h_2, \cdots,h_{L-1}]\) where \(h_0\) is the CIR at symbol sampling instant \(0T_{sym}\) and \(h_{L-1}\) is the CIR at symbol sampling instant \((L-1)T_{sym}\). Such a channel can be modeled as a tapped delay line (TDL) filter, otherwise called finite impulse response (FIR) filter. Here, we only consider the CIR at symbol sampling instances. It is well known that the output of such a channel (\(\mathbf{r}\)) is given as the linear convolution of the input symbols (\(\mathbf{s}\)) and the CIR (\(\mathbf{h}\)) at symbol sampling instances. In addition, channel noise in the form of AWGN can also be included the model. Therefore, the resulting vector of from the entire channel model is given as

\[\mathbf{r} = \mathbf{h} \ast \mathbf{s} +\mathbf{n} \quad\quad (1) \]

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Simulation model for detection in flat fading channel

A flat-fading (also called as frequency-non-selective) channel is modeled with a single tap (\(L=1\)) FIR filter with the tap weights drawn from distributions like Rayleigh, Rician or Nakagami distributions. We will assume block fading, which implies that the fading process is approximately constant for a given transmission interval. For block fading, the random tap coefficient \(h=h[0]\) is a complex random variable (not random processes) and for each channel realization, a new set of complex random values are drawn from Rayleigh or Rician or Nakagami fading according to the type of fading desired.

Figure 2: LTI channel viewed as tapped delay line filter

Simulation models for modulation and detection over a fading channel is shown in Figure 2. For a flat fading channel, the output of the channel can be expressed simply as the product of time varying channel response and the input signal. Thus, equation (1) can be simplified (refer this article for derivation) as follows for the flat fading channel.

\[\mathbf{r} = h\mathbf{s} + \mathbf{n} \quad\quad (2) \]

Since the channel and noise are modeled as a complex vectors, the detection of \(\mathbf{s}\) from the received signal is an estimation problem in the complex vector space.

Figure 3: Simulation model for modulation and detection over flat fading channel

Assuming perfect channel knowledge at the receiver and coherent detection, the receiver shown in Figure 3(a) performs matched filtering. The impulse response of the matched filter is matched to the impulse response of the flat-fading channel as \( h^{\ast}\). The output of the matched filter is scaled down by a factor of \(||h||^2\) which is the total-energy contained in the impulse response of the flat-fading channel. The resulting decision vector \(\mathbf{y}\) serves as the sufficient statistic for the estimation of \(\mathbf{s}\) from the received signal \(\mathbf{r}\) (refer equation A.77 in reference [1])

\[\tilde{\mathbf{y}} = \frac{h^{\ast}}{||h||^2} \mathbf{r} = \frac{h^{\ast}}{||h||^2} h\mathbf{s} + \frac{h^{\ast}}{||h||^2} \mathbf{n} = \mathbf{s} + \tilde{\mathbf{w}} \quad\quad (3) \]

Since the absolute value \(|h|\) and the Eucliden norm \(||h||\) are related as \(|h|^2= \left\lVert h\right\rVert = hh^{\ast}\), the model can be simplified further as given in Figure 3(b).

To simulate flat fading, the values for the fading variable \(h\) are drawn from complex normal distribution

\[h= X + jY \quad\quad (4) \]

where, \(X,Y\) are statistically independent real valued normal random variables.

● If \(E[h]=0\), then \(|h|\) is Rayleigh distributed, resulting in a Rayleigh flat fading channel
● If \(E[h] \neq 0\), then \(|h|\) is Rician distributed, resulting in a Rician flat fading channel with the factor \(K=[E[h]]^2/\sigma^2_h\)

References

[1] D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005.↗

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