Module-16:Finite Impulse Response (FIR) Digital Filters Objectives: • To understand the concept of linear phase • To know the method of identifying the linear phase filter from its impulse response • To know the ways of identifying various types of linear phase filters • To understand the characteristics of FIR digital filters • To understand the frequency response of FIR filters • To differentiate between FIR and IIR filters based on various Characteristics Introduction: ❖ Linear Time Invariant systems can be classified according to whether the unit sample response sequence is of finite duration or is of infinite duration i.e. whether or not it has only a finite number of non zero terms. ❖ If the unit sample response sequence of the LTI system is of finite duration, the system is referred to as Finite Impulse Response system. ❖ The system with unit sample response ℎ(푛) = 푎푛푢(푛) is an example for IIR system and the system with unit sample response ℎ(푛) = 2 푓표푟 푛 = −1, 1 { 1 푓표푟 푛 = 0 } is an example for FIR system. 0 표푡ℎ푒푟푤푖푠푒 ∞ ❖ The response 푦(푛) for an input of 푥(푛) has the form 푦(푛) = ∑푘=−∞ 푥(푛 − 푘)ℎ(푘) = 푥(푛 + 1)ℎ(−1) + 푥(푛)ℎ(0) + 푥(푛 − 1)ℎ(1) = 2푥(푛 + 1) + 푥(푛) + 2푥(푛 − 1) This is the difference equation or recurrence relation expressing 푦(푛) in terms of the input only. It does not depend on other output values. ❖ A discrete LTI system can be described by the Nth order linear constant coefficient difference equation as 푁 푀 ∑푘=0 푎푘 푦(푛 − 푘) = ∑푚=0 푏푚 푥(푛 − 푚) ----(1) 1 푁 푀 ❖ From the above eqn., 푦(푛) = [− ∑푘=1 푎푘 푦(푛 − 푘) + ∑푚=0 푏푚 푥(푛 − 푎0 푚)] ---(2) ❖ If some 푎푘, 1 ≤ 푘 ≤ 푁 푖푠 푛표푛 푧푒푟표, the system represented by above eqn. Is an IIR system. ❖ If some 푎푘 = 0, 1 ≤ 푘 ≤ 푁, the above eqn. becomes 푦(푛) = 푀 푏푚 ∑푚=0 푥(푛 − 푚) --(3) 푎0 ∞ ❖ Comparing this eqn. with 푦(푛) = ∑푚=−∞ ℎ(푚)푥(푛 − 푚), 푏푚 푓표푟 푚 = 0,1, … 푀 ℎ(푚) = [ 푎0 ] which is an FIR system ---(4) 0 표푡ℎ푒푟푤푖푠푒 ❖ A discrete LTI is stable if and only if its unit sample response is absolutely summable i.e. ∞ ∑푘=−∞|ℎ(푘)| < ∞ ---(5) ❖ An FIR system described by eqn.(1) when 푎푘 = 0, 푘 = 1,2, … , 푁 is always stable since there are only a finite number of non zero samples h(n) and the series in eqn.(5) always converges. ❖ For any system to be physically realizable, it should be causal. ❖ A LTI system can be checked for causality by using the condition ℎ(푛) = 0 푓표푟 푛 < 0 , where h(n) is the unit sample response of the system. ❖ In the case of an FIR filter, even though h(n) is non causal, it can be sufficiently delayed to obtain a causal sequence( because of its finite length) representing a realizable filter. Thus, a realizable filter can always be obtained for an FIR filter. concept of Linear Phase: ➢ An ideal filter shows perfect transmission in the pass band and perfect ejection in the stop band ➢ Perfect transmission implies a transfer function with constant gain and linear phase over the pass band. ➢ The corresponding transfer Function can be given as 퐻(푓) = 퐾. 푒−푗푛휔, over the pass band, where 휔 = 2휋푓 ➢ The gain of a filter is given by |퐻(푓)| ➢ Constant gain 퐾 ensures that the output is an amplitude scaled replica of the input. ➢ If the gain is not constant over the required frequency range, it results in amplitude distortion ➢ The phase response of the filter is 휑(휔) = −휔푛. Linear phase ensures that the output is a time shifted replica of the input. ➢ If the phase shift is not linear with frequency, it results in Phase distortion, as the signal undergoes different delays for different frequencies. ➢ The DC gain of the filter will be at 푓 = 0 표푟 휔 = 0 and the high frequency gain is at 푓 = 0.5 표푟 휔 = 휋 . ➢ Computation of DC gain: ▪ 퐻(푧)푎푡 푧 = 1 ▪ 퐻(휔)푎푡 푓 = 0 표푟 휔 = 0 ▪ 퐻(0) = ∑ ℎ(푛) ▪ Ratio of sum of the RHS coefficients to that of LHS coefficients of the difference equation, where RHS refers to excitation and LHS refers to the response ➢ Computation of High Frequency gain: ▪ 퐻(푧)푎푡 푧 = −1 ▪ 퐻(휔) 푎푡 푓 = 0.5 표푟 휔 = 휋 푛 ▪ 퐻(휔)|휔=0.5 = ∑(−1) ℎ(푛) ▪ 푅푒푣푒푟푠푒 푡ℎ푒 푠푖푔푛 표푓 푎푙푡푒푟푛푎푡푒 푐표푒푓푓푖푐푖푒푛푡푠 표푓 푅퐻푆 푎푛푑 퐿퐻푆 표푓 푡ℎ푒 푑푖푓푓푒푟푒푛푐푒 푒푞푢푎푡푖표푛 푎푛푑 푡푎푘푒 푡ℎ푒 푟푎푡푖표 표푓 푡ℎ푒푖푟 푠푢푚 ➢ Phase delay of a sinusoid 푥(푡) = 퐴. cos(휔0푡 + 휃) = 퐴. cos [휔0(푡 − 푡푝)] is the quantity 휃 푡푝 = − and describes the time delay in the signal caused by a phase shift 휔0 of ‘휃′ 1 퐻(푓0) ➢ Phase delay of a digital filter is defined as 푡푝(푓0) = − . 2휋 푓0 ➢ Ex. 푐표푛푠푖푑푒푟 푦(푛) = 푥(푛) + 푥(푛 − 1) 푗휔 2 휔 − ⁄ . 푇ℎ푒 푐표푟푟푒푠푝표푛푑푖푛푔 퐻(휔) = 4퐶표푠 ( ) . 푒 2. 2 푇ℎ푢푠, 푡ℎ푒 푝ℎ푎푠푒 푟푒푠푝표푛푠푒 푖푠휃(휔) = 휔 휃(휔) 1 − . 푇ℎ푒 푐표푟푟푒푠푝표푛푑푖푛푔 푝ℎ푎푠푒 푑푒푙푎푦 푖푠 = . This system exhibits 2 휔 2 half a sample of time delay at each frequency. 1 푑퐻(푓) ➢ The Group delay of a digital filter is defined as푡 (푓) = − . 푔 2휋 푑푓 ➢ The group delay may be interpreted as the time delay of the amplitude of a sinusoid at frequency 휔 ➢ It refers to the fact that it specifies the delay experienced by a narrow-band ``group'' of sinusoidal components which have frequencies within a narrow frequency interval about ➢ For linear phase responses, i.e.,휃(휔) = −훼휔 for some constant 훼, the group delay and the phase delay are identical, and each may be interpreted as time delay (equal to 훼 samples when 휔 ∈ (−휋, 휋) ➢ For the system 푦(푛) = 푥(푛) + 푥(푛 − 1), 푡ℎ푒 푝ℎ푎푠푒 푟푒푠푝표푛푠푒 푖푠휃(휔) = 휔 − . 2 ➢ Thus, both the phase delay and the group delay of the above system are equal to half a sample at every frequency. ➢ For a linear phase filter, both Group delay and phase delay are constant. ➢ If the phase response is nonlinear then the relative phases of the sinusoidal signal components are generally altered by the filter. FIR filters and Linear Phase: ▪ The DTFT of a filter whose impulse response is symmetric about the origin is purely real or purely imaginary. ▪ The phase of such a filter is thus piecewise constant ▪ If the symmetric sequence is shifted (to make it causal), the phase is augmented by a linear phase term and becomes piecewise linear ▪ Generally, a filter whose impulse response is symmetric about its midpoint is termed as Linear Phase Filter. ▪ Linearity in phase results in a pure time delay with no amplitude distortion. ▪ An FIR filter with an impulse response symmetric about the midpoint will have linear phase and provides constant delay. ▪ For the sequence h(n) to be of finite length, the poles must lie at the origin. ▪ Sequences that are symmetric about the origin also require ℎ(푛) = ±ℎ(푛) 1 , which implies 퐻(푧) = ±퐻( ) 푧 ▪ The zeros of a linear phase sequence must occur in reciprocal pairs(conjugate pairs if complex to ensure real coefficients) Classification of Linear Phase Sequences ➢ The length N of finite symmetric sequences can be odd or even, since the center of symmetry may fall on a sample point (for odd N) or midway between samples (for even N). This results in four possible types of symmetric sequences Type-1 Sequence: • It has even symmetry and Odd length N and the centre of symmetry being 푁−1 the integer value = . 2 푁−1 −푗휔( ) • It’s frequency response may be expressed as 퐻(푓) = 퐻1(푓). 푒 2 , where 푁−3 ( ) 2 푁 − 1 푁 − 1 퐻 (푓) = ℎ ( ) + 2 ∑ ℎ(푛). cos (n − ) 휔 1 2 2 푛=0 푁−1 푁−1 • It has a linear phase of – 휔( ) and a constant group delay of . 2 2 • The amplitude spectrum is even symmetric about both f = 0 and f = 1 when the spectrum is studied over = (−2휋 ,2휋) . |퐻1(0)|푎푛푑|퐻1(1)| can be non zero . • The amplitude spectrum is even symmetric about both f=0 and f=0.5, when the spectrum is studied over = (−휋 ,휋) . |퐻1(0)|푎푛푑|퐻1(0.5)| can be non zero **It is sufficient to study over (ퟎ, ퟐ흅) 풐풓 (−흅, 흅) Type-2 Sequence • It has even symmetry and even length N and the centre of symmetry being 푁−1 the half integer value = 2 푁−1 −푗휔( ) • It’s frequency response may be expressed as 퐻(푓) = 퐻1(푓). 푒 2 , where 푁 ( −1) 2 푁 − 1 퐻 (푓) = 2 ∑ ℎ(푛). cos (n − ) 휔 1 2 푛=0 푁−1 푁−1 • It has a linear phase of – 휔( ) and a constant group delay of . 2 2 • The amplitude spectrum has even symmetry about f = 0 and odd symmetry about f = 1 when the spectrum is studied over 휔 = (−2휋 ,2휋).As a result, |퐻1(1)| is always zero . • The amplitude spectrum is even symmetric about both f=0 and f=0.5, when the spectrum is studied over = (−휋 ,휋).|퐻1(0.5)| is always zero Type-3 Sequence • It has odd symmetry and odd length N and the centre of symmetry being 푁−1 the integer value = 2 푁−1 휋 −푗휔( ) 푗 • It’s frequency response may be expressed as 퐻(푓) = 퐻1(푓).