The Discrete-Time Fourier

Chapter 3 The Discrete-Time Fourier Transform 清大電機系林嘉文 cwlin@ee.nthu.edu.tw 03-5731152 Original PowerPoint slides prepared by S. K. Mitra 3-1-1

Continuous-Time Fourier Transform Definition The CTFT of a continuous-time esg signal x a (t) is given by Often referred to as the Fourier spectrum or simply the spectrum of the continuous-timetime signal Definition The inverse CTFT of a Fourier transform X a ( jω) is given by a( j ) g y Often referred to as the Fourier integral A CTFT pair will be denoted as Original PowerPoint slides prepared by S. K. Mitra 3-1-2

Continuous-Time Fourier Transform Ω is real and denotes the continuous-time angular frequency variable in radians In general, the CTFT is a complex function of Ω in the range < Ω < It can be expressed in the polar form as where The quantity ǀX a ( jω)ǀ is called the magnitude spectrum and the quantity θ a (Ω) is called the phase spectrum Both spectrums are real functions of Ω In general, the CTFT X a ( jω) exists if x a (t) satisfies the Dirichlet conditions Original PowerPoint slides prepared by S. K. Mitra 3-1-3

Dirichlet Conditions (a) The signal x a (t) has a finite number of discontinuities t and a finite number of maxima and minima in any finite interval (b) The signal is absolutely integrable, i.e., x a () t dt < If the Dirichlet conditions are satisfied, then converges to x a(t) except at values of t where x a(t) has discontinuities It can be shown that if x a (t) is absolutely integrable, then ǀX a ( jω)ǀ < proving the existence of the CTFT Original PowerPoint slides prepared by S. K. Mitra 3-1-4

Energy Density Spectrum The total energy E x of a finite energy continuous-time complex signal x a (t) is given by which can also be rewritten as Interchanging the order of the integration we get Original PowerPoint slides prepared by S. K. Mitra 3-1-5

Hence Energy Density Spectrum This is commonly known as the Parseval s relation for finite-energy continuous-time signals The quantity ǀX a ( jω)ǀ 2 is called the energy density spectrum of x a (t) and usually denoted as The energy over a specified range of Ω a Ω Ω b can be computed using Original PowerPoint slides prepared by S. K. Mitra 3-1-6

Band-limited Continuous-Time Signals (1/2) A full-band, finite-energy, continuous-time signal has a spectrum occupying the whole frequency range <Ω< Ab band-limited dli it d continuous-time signal has a spectrum that t is limited to a portion of the frequency range <Ω< An ideal band-limited signal has a spectrum that is zero outside a finite frequency range Ω a ǀΩǀ Ω b, that is However, an ideal band-limited signal cannot be generated in practice Original PowerPoint slides prepared by S. K. Mitra 3-1-7

Band-limited Continuous-Time Signals (2/2) Band-limited signals are classified according to the frequency range where most of the signal s energy is concentrated A lowpass, continuous-time signal has a spectrum occupying the frequency range ǀΩǀ Ω p <, where Ω p is called the bandwidth of the signal A highpass, continuous-time signal has a spectrum occupying the frequency range 0 < Ω p ǀΩǀ < where the bandwidth of the signal is from Ω p to A bandpass, continuous-time signal has a spectrum occupying the frequency range 0 < Ω L ǀΩǀ Ω H <, where Ω H Ω L is the bandwidth Original PowerPoint slides prepared by S. K. Mitra 3-1-8

Discrete-Time Fourier Transform Definition - The discrete-time time Fourier transform (DTFT) X (e jω ) of a sequence x[n] is given by In general, X(e jω ) is a complex function of ω as follows X re (e jω ) and X im (e jω ) are, respectively, the real and imaginary parts of X(e jω ), and are real functions of ω X(e jω ) can alternately be expressed as where Original PowerPoint slides prepared by S. K. Mitra 3-1-9

Discrete-Time Fourier Transform ǀX(e jω )ǀ is called the magnitude function θ(ω) is called the phase function In many applications, the DTFT is called the Fourier spectrum Likewise, ǀX(e jω )ǀ and θ(ω) are called the magnitude and phase spectra For a real sequence x[n], [ ǀX(e( jω )ǀ and X re( (e jω ) are even functions of ω, whereas, θ(ω) and X im (e jω ) are odd functions of ω jω jω j( θ ( ω) + 2πk ) jω jθ ( ω ) Note: X ( e ) = X ( e ) e = X ( e ) e The phase function θ(ω) cannot be uniquely specified for any DTFT Original PowerPoint slides prepared by S. K. Mitra 3-1-10

Discrete-Time Fourier Transform If not specified, ed, we shall assume that the phase function θ(ω) is restricted to the following range of values: π θ(ω) ( ) <π called the principal value The DTFTs of some sequences exhibit discontinuities of 2π in their phase responses An alternate type of phase function that is a continuous function of ω can be derived from the original phase function by removing the discontinuities of 2π, denoted by θ c (ω) The process of removing the discontinuities is called unwrapping Original PowerPoint slides prepared by S. K. Mitra 3-1-11

Discrete-Time Fourier Transform Example The DTFT of unit sample sequence δ[n] is given by Example Consider the causal sequence Its DTFT is given by X ( jω ) n jωn n jωn e α μ[ n e = α e = ] n= n= 0 1 αe ( jω α ) n e = jω = 1 α jω n= 0 as αe jω = α < 1 Original PowerPoint slides prepared by S. K. Mitra 3-1-12

Discrete-Time Fourier Transform The magnitude and phase of the DTFT X (e jω ) =1/(1 0.5e jω ) are shown below Original PowerPoint slides prepared by S. K. Mitra 3-1-13

Discrete-Time Fourier Transform The DTFT X(e jω )of x[n] is a continuous function of ω It is also a periodic function of ω with a period 2π: Therefore represents the Fourier series representation of the periodic function As a result, the Fourier coefficients x[n] can be computed from using the Fourier integral Original PowerPoint slides prepared by S. K. Mitra 3-1-14

Discrete-Time Fourier Transform Inverse discrete-time Fourier transform: Proof: The order of integration and summation can be interchanged if the summation inside the brackets converges uniformly, i.e. X(e jω ) exists Then Original PowerPoint slides prepared by S. K. Mitra 3-1-15

Discrete-Time Fourier Transform Now Hence Original PowerPoint slides prepared by S. K. Mitra 3-1-16

Discrete-Time Fourier Transform Convergence Condition - An infinite series of the form may or may not converge Let Then for uniform convergence of X(e jω ) Now, if x[n] is an absolutely summable sequence, i.e., if Original PowerPoint slides prepared by S. K. Mitra 3-1-17

Discrete-Time Fourier Transform Then for all values of ω Thus, the absolute summability of x[n] is a sufficient condition for the existence of the DTFT X(e jω ) Original PowerPoint slides prepared by S. K. Mitra 3-1-18

Discrete-Time Fourier Transform Example the sequence x[n] =αα n μ[n] for α <1is absolutely summable as n n 1 α μ n = α = n= n= 0 1 [ ] < and its DTFT X(e jω ) therefore converges to 1/(1 αe jω ) uniformly Since 2 2 x[ n] x[ n] n= n= an absolutely summable sequence has always a finite energy However, a finite-energy sequence is not necessarily absolutely summable Original PowerPoint slides prepared by S. K. Mitra 3-1-19 α

Discrete-Time Fourier Transform Example the sequence has a finite it energy equal to But, x[n] is not absolutely summable To represent a finite it energy sequence x[n] [ ] that t is not absolutely summable by a DTFT X(e jω ), it is necessary to consider mean square convergence of X(e jω ) where lim π K π X ( jω ) ( jω e X e ) dω = 0 Original PowerPoint slides prepared by S. K. Mitra 3-1-20 K 2

Discrete-Time Fourier Transform Here, the total energy of the error must approach zero at each value of ω as K goes to In such a case, the absolute value of the error X(e jω ) X jω K (e ) may not go to zero as K goes to and the DTFT is no longer bounded Example Consider the following DTFT: Original PowerPoint slides prepared by S. K. Mitra 3-1-21

Discrete-Time Fourier Transform The inverse DTFT of H LP(e jω ) is sgiven by The energy of h LP [n] is given by ω c / π h LP [n] is a finite-energy sequence, but it is not absolutely summable As a result does not uniformly converge to H LP (e jω ) for all values of ω, but converges to H LP (e jω ) in the mean-square sense Original PowerPoint slides prepared by S. K. Mitra 3-1-22

Discrete-Time Fourier Transform The mean-square convergence property of the sequence h LP [n] can be further illustrated by examining the plot of the function Original PowerPoint slides prepared by S. K. Mitra 3-1-23

Discrete-Time Fourier Transform As can be seen from these plots, independent of the value of K there are ripples in the plot of H LP,K (e jω ) around both sides of the point ω = ω c The number of ripples increases as K increases with the height of the largest ripple remaining the same for all values of K As K goes to infinity, the condition holds indicating the convergence of H jω jω LP,K (e ) to H LP (e ) The oscillatory behavior of H LP,K (e jω ) approximating H LP( (e jω ) in the mean-square sense at a point of discontinuity is known as the Gibbs phenomenon Original PowerPoint slides prepared by S. K. Mitra 3-1-24

Discrete-Time Fourier Transform The DTFT can also be defined ed for a certain class of sequences which are neither absolutely summable nor square summable e.g., the unit step sequence μ[n], the sinusoidal sequence cos(ω o n + φ), and the exponential sequence Aα n For this type of sequences, a DTFT representation is possible using the Dirac delta function δ(ω), a function of ω with infinite height, eg zero width, and unit area It is the limiting form of a unit area pulse function p (ω) as goes to zero satisfying Original PowerPoint slides prepared by S. K. Mitra 3-1-25

Discrete-Time Fourier Transform Example Consider the following exponential sequence Its DTFT is given by where δ(ω) is an impulse function of ω and π ω o π The function is a periodic function of ω with a period 2π and is called a periodic impulse train Original PowerPoint slides prepared by S. K. Mitra 3-1-26

Discrete-Time Fourier Transform To verify that X(e jω ) given above is indeed the DTFT of 0 x[n] = we compute the inverse DTFT of X(e jω ) Thus j n e ω where we have used the sampling property p of the impulse function δ(ω) Original PowerPoint slides prepared by S. K. Mitra 3-1-27

Commonly Used DTFT Pairs Sequence DTFT u[n] α n. Original PowerPoint slides prepared by S. K. Mitra 3-1-28

DTFT Properties: Symmetry Relations (Complex Sequences) Original PowerPoint slides prepared by S. K. Mitra 3-1-29

DTFT Properties: Symmetry Relations (Real Sequences) Original PowerPoint slides prepared by S. K. Mitra 3-1-30

General Properties of DTFT Original PowerPoint slides prepared by S. K. Mitra 3-1-31

DTFT Properties Example Determine the DTFT Y(e jω )ofy[n] Let x[n] =αα n μ[n], α <1 We can therefore write y[n] [ ] = nx[n] [ ]+ x[n] [ ] From Table 3.3, the DTFT of x[n] is given by Using the differentiation property of DTFT (Table 3.2), we observe that the DTFT of nx[n] is given by Original PowerPoint slides prepared by S. K. Mitra 3-2-32

DTFT Properties Next using the linearity property DTFT (Table 3.4) we arrive at Example Determine the DTFT V(e( jω ) of v[n] [ ] d 0 v[n] + d 1 v[n 1] = p 0 δ[n] + p 1 δ[n 1] Using the time-shifting gproperty p of DTFT (Table 3.4) we observe that the DTFT of δ[n 1] is e jω and the DTFT of v[n 1] is e jω V(e jω ) Using the linearity property we then obtain the frequency-domain representation of Original PowerPoint slides prepared by S. K. Mitra 3-2-33

Energy Density Spectrum The total energy of a finite-energy sequence g[n] is given by From Parseval s relation we observe that The following quantity is called the energy density spectrum The area under this curve in the range π ω π divided by 2π is the energy of the sequence Original PowerPoint slides prepared by S. K. Mitra 3-2-34

Energy Density Spectrum Example Compute the energy of the sequence Here where Therefore Hence h LP[ [n] is a finite-energy lowpass sequence Original PowerPoint slides prepared by S. K. Mitra 3-2-35

DTFT Computation Using MATLAB (1/2) The function freqz can be used to compute the values of the DTFT of a sequence, described as a rational function in the form of Usage: H = freqz(num,den,w) The function returns the frequency response values as a vector H of a DTFT defined in terms of the vectors num and den containing the coefficients {p i } and {d i }, respectively at a prescribed set of frequencies between 0 and 2π given by the vector w Original PowerPoint slides prepared by S. K. Mitra 3-2-36

DTFT Computation Using MATLAB (2/2) jω X( e ) = 0.008 0.033e + 0.05e 0.033e + 0.008e jω j2ω j3ω j4ω 1 + 237 2.37 e + 27 2.7 e + 16 1.6 e + 0.41 041 e jω j2ω j3ω j4ω Original PowerPoint slides prepared by S. K. Mitra 3-2-37

Linear Convolution Using DTFT The convolution theorem states that if y[n] = x[n] then the DTFT Y(e jω ) of y[n] is given by Y(e jω ) = X(e jω )H(e jω ) h[n], An implication of this result is that the linear convolution y[n] of x[n] and h[n] can be performed as follows: 1) Compute the DTFTs X(e jω ) and H(e jω ) of the sequences x[n] and h[n], respectively 2) Compute Y(e jω ) = X(e jω )H(e jω ) 3) Compute the IDFT y[n] of Y(e jω ) Original PowerPoint slides prepared by S. K. Mitra 3-2-38

Unwrapping the Phase (1/2) In numerical computation, when the computed phase function is outside the range [ π,π], the phase is computed modulo 2π, to bring the computed value to [ π,π] Thus, the phase functions of some sequences exhibit discontinuities of radians in the plot jω X( e ) = 0.008 0.033e + 0.05e 0.033e + 0.008e jω j2ω j3ω j4ω 1+ 2.37e + 2.7e + 1.6e + 0.41e jω j2ω j3ω j4ω Original PowerPoint slides prepared by S. K. Mitra 3-2-39

Unwrapping the Phase (2/2) In such cases, often an alternate type of phase function that is continuous function of ω is derived from the original function by removing the discontinuities of 2π Process of discontinuity removal is called unwrapping the phase The unwrapped phase function will be denoted d θ c (ω) In MATLAB, the unwrapping can be implemented using unwrap Original PowerPoint slides prepared by S. K. Mitra 3-2-40

The Frequency Response (1/6) Most discrete-time signals encountered in practice can be represented as a linear combination of a very large, maybe infinite, number of sinusoidal discrete-time signals of different angular frequencies Thus, knowing the response of the LTI system to a single sinusoidal signal, we can determine its response to more complicated signals by making use of the superposition property An important property of an LTI system is that for certain types of input signals, called eigen functions, the output signal is the input signal multiplied by a complex constant Original PowerPoint slides prepared by S. K. Mitra 3-2-41

The Frequency Response (2/6) Consider the LTI discrete-time t e system with an impulse response {h[n]} shown below Its input-output relationship in the time-domain is given by the convolution sum If the input is of the form x[n] = e jωn, < n < then it follows that the output is given by Original PowerPoint slides prepared by S. K. Mitra 3-2-42

Let The Frequency Response (3/6) Then we can write y[n] = H(e jω )e jωn Thus for a complex exponential input signal e jωn, the output t of an LTI discrete-time ti system is also a complex exponential signal of the same frequency multiplied by a complex constant H(e jω ) Thus e jωn is an eigen function of the system The quantity H(e jω ) is called the frequency response of the LTI discrete-time system H(e jω ) provides a frequency-domain description of the system Original PowerPoint slides prepared by S. K. Mitra 3-2-43

The Frequency Response (4/6) H(e jω ), in general, is a complex function of ω with a period 2π, with its real and imaginary parts as follows: or, in terms of its magnitude and phase, where θ(ω) = arg H(e jω ) The function H(e jω ) is called the magnitude response and the function θ(ω) is called the phase response of the LTI discrete-timetime system Design specifications for the LTI discrete-time system, in many applications, are given in terms of the magnitude response or the phase response or both Original PowerPoint slides prepared by S. K. Mitra 3-2-44

The Frequency Response (5/6) In some cases, the magnitude function is specified ed in decibels as jω G ω ) = 20log H ( e ) db ( 10 where G(ω) is called the gain function The negative of the gain function A(ω) = G(ω) is called the attenuation or loss function If the impulse response h[n] is real then the magnitude function is an even function of ω H(e jω ) = H(e jω ) and the phase function is an odd function of ω: θ(ω) = θ( ω) Original PowerPoint slides prepared by S. K. Mitra 3-2-45

The Frequency Response (6/6) Likewise, for a real impulse response se h[n], [ H re(e jω ) is even and H im (e jω ) is odd Example - M-point p moving average filter with an impulse response given by Its frequency response is then given by Or, H e 1 1 e e e M n= 0 n= M M n= 0 e jmω 1 1 e 1 sin ( Mω / 2) jm ( 1 ) ω /2 = = e jω M 1 e M sin ω /2 jω jωn jωn jωn jmω ( ) = = ( 1 ) ( ) Original PowerPoint slides prepared by S. K. Mitra 3-2-46

Computing Frequency Response Using MATLAB The function freqz(h,1,w) can be used to determine the values of the frequency response vector h at a set of given frequency points w From h, the real and imaginary parts can be computed using the functions real and imag, and the magnitude and phase functions using the functions abs and angle M-point moving average filter Original PowerPoint slides prepared by S. K. Mitra 3-2-47

Steady State Response (1/3) Note that the frequency response also determines the steady-state response of an LTI discrete-time system to a sinusoidal input Example Determine the steady-state state output t y[n] of a real coefficient LTI discrete-time system with a frequency response H(e jω ) for an input x[n] = Acos(ω o n + ϕ), < n < We can express the input x[n] as x[n] = g[n] + g*[n] where j o Now the output of the system for an input e ω n is simply Original PowerPoint slides prepared by S. K. Mitra 3-2-48

Steady State Response (2/3) Because of linearity, the response se v[n] [ ] to an input g[n] ] is given by Because of linearity, the response v*[n] to an input g*[n] is given by Combining the last two equations we get Original PowerPoint slides prepared by S. K. Mitra 3-2-49

Steady State Response (2/3) Thus, the output y[n] has the same sinusoidal waveform as the input with two differences: 1) the amplitude is multiplied by e jω 0 H the value of the magnitude function at ω = ω o ( ) 2) the output has a phase lag relative to the input by an amount θ(ω o ), the value the value of the phase function at ω = ω o Original PowerPoint slides prepared by S. K. Mitra 3-2-50

Response to a Causal Exponential Sequence The expression for the steady-state response developed earlier assumes that the system is initially relaxed before the application of the input x[n] [ ] In practice, excitation x[n] to a system is usually a rightsided sequence applied at some sample index n = n o Without any loss of generality, assume x[n] = 0 for n < 0 From the input-output relation we observe that for an input k= y [ n] = h[ k] x[ n k] x[n] = e jωn μ[n] n the output is given by j ω ( n k ) y [ n ] = h [ k ] e μ [ n ] k = 0 Original PowerPoint slides prepared by S. K. Mitra 3-2-51

Response to a Causal Exponential Or, n y [ n ] = h [ k ] e k= 0 Sequence j ω k e j ω n μ [ n ] The output for n <0is y[n] =0 The output for n 0 is given by n jωk jωn jωk jωn jωk jωn y[ n] = h[ k] e e = h[ k] e e h[ k] e e k= 0 k= 0 k= n+ 1 Or, ( jω ) jωn jωk jωn y[ n] = H e e h[ k] e e k = n+ 1 The first term on the RHS is the same as that obtained when the input is applied at n = 0 to an initially relaxed system and is the steady-state response: Original PowerPoint slides prepared by S. K. Mitra 3-2-52

Response to a Causal Exponential Sequence The second term on the RHS is called the transient response: To determine the effect of the above term on the total output response, we observe For a causal, stable LTI IIR discrete-time system, h[n] is absolutely summable As a result, the transient response y tr [n] is a bounded sequence Moreover, as n Original PowerPoint slides prepared by S. K. Mitra 3-2-53

Response to a Causal Exponential Sequence For a causal FIR LTI discrete-time system with an impulse response h[n], of length N + 1, h[n] = 0 for n > N Hence, y tr [n] = 0 n > N 1 Here the output reaches the steady-state value y sr [n] = H(e jω ) e jω at n = N Original PowerPoint slides prepared by S. K. Mitra 3-2-54

The Concept of Filtering (1/8) Filtering is to pass certain frequency components in an input sequence without any distortion (if possible) while blocking other frequency components The key to the filtering process is It expresses an arbitrary input as a linear weighted sum of an infinite number of exponential/sinusoidal sequences Thus, by appropriately choosing the values of H(e jω ) of the filter at concerned frequencies, some of these components can be selectively heavily attenuated or filtered with respect to the others Original PowerPoint slides prepared by S. K. Mitra 3-2-55

The Concept of Filtering (2/8) Consider a real-coefficient LTI discrete-time system characterized by a magnitude function We apply the following input to the system x[n] =Acosω 1 n+bcosω 2 n, 0<ω ω 1 < ω c < ω 2 < π Because of linearity, the output of this system is of the form As jω1 jω2 ( ) H ( e ) H e the output reduces to 1 0 (lowpass filter) Original PowerPoint slides prepared by S. K. Mitra 3-2-56

The Concept of Filtering (3/8) Example - The input consists of two sinusoidal sequences of frequencies 0.1 rad/sample and 0.4 rad/sample We need to design a highpass filter that will only pass the high-frequency enc component of the input Assume the filter to be an FIR filter of length 3 with an impulse response: h[0] = h[2] = α, h[1] = β The convolution sum description of this filter is given by y[n] = h[0]x[n] + h[1]x[n 1] + h[2]x[n 2] = αx[n] + βx[n 1] + αx[n 2] Design Objective: Choose suitable values of α and β so that the output is a sinusoidal sequence with a frequency 0.4 rad/sample Original PowerPoint slides prepared by S. K. Mitra 3-2-57

The Concept of Filtering (4/8) The frequency response of the FIR filter is given by The magnitude and phase functions are H(e jω ) = 2α cosω + β θ(ω) = ω To block the low-frequency component and pass the high-frequency one, the magnitude function at ω = 0.1 should be equal to zero, while that at ω = 0.4 should be equal to one Original PowerPoint slides prepared by S. K. Mitra 3-2-58

The Concept of Filtering (5/8) Thus, the two conditions o that must be satisfied s are H(e j0.1 ) = 2αcos(0.1) + β = 0 H(e j0.4 4) = 2αcos(0.4) + β = 0 Solving the above two equations we get α = 6.76195 β =13.456335 Thus the output-input relation of the FIR filter is given by y[n] = 6.76195(x[n] 6 + x[n 2]) + 13.456335x[n 1] where the input is x[n] = {cos(0.1n) + cos(0.4n)}μ[n] Original PowerPoint slides prepared by S. K. Mitra 3-2-59

The Concept of Filtering (6/8) The waveforms of input and output signals are shown below Original PowerPoint slides prepared by S. K. Mitra 3-2-60

The Concept of Filtering (7/8) The first seven samples of the output are shown below It can be seen that, neglecting the least significant digit y[n] = cos(0.4(n 1)) for n 2 Computation of the present output value requires the knowledge of the present and two previous input samples Original PowerPoint slides prepared by S. K. Mitra 3-2-61

The Concept of Filtering (8/8) Hence, the first two output samples, y[0] and y[1], are the result of assumed zero input sample values at n = 1 and n = 2 Therefore, first two output samples constitute the transient part of the output Since the impulse response is of length 3, the steadystate is reached at n = N = 2 Note also that the output is delayed version of the highfrequency component cos(0.4n) of the input, and the delay is one sample period Original PowerPoint slides prepared by S. K. Mitra 3-2-62

Phase Delay If the input x[n] [ ] to an LTI system H(e jω ) is a sinusoidal signal of frequency ω o x[n] [ ] = Acos(ω on + ϕ), < n < Then, the output y[n] is also a sinusoidal signal of the same frequency ω o but lagging in phase by θ(ω o ) radians: x[n] = A H(e jω ) cos(ω o n + θ(ω o ) + ϕ), < n < We can rewrite the output expression as x[n] = A H(e jω ) cos(ω o ( n τ p (ω o ) + ϕ)), < n < where τ p (ω o ) = θ(ω o ) / ω o is called the phase delay The minus sign in front indicates phase lag In general, y[n] will not be a delayed replica of x[n] unless the phase delay is an integer Original PowerPoint slides prepared by S. K. Mitra 3-2-63

Group Delay When the input is composed of several e sinusoidal components with different frequencies that are not harmonically related, each component will go through different phase delays In this case, the signal delay is determined using the group delay defined d by In defining the group delay, it is assumed that the phase function is unwrapped so that its derivatives exist Group delay has a physical meaning only with respect to the underlying continuous-time functions associated with y[n] and x[n] Original PowerPoint slides prepared by S. K. Mitra 3-2-64

Phase and Group Delay A graphical comparison of the two types of delays: Example - The phase function of the FIR filter y[n] = αx[n] + βx[n 1] + αx[n 2] is θ(ω) = ω Hence its group delay τ g (ω) is given by verifying the result obtained earlier by simulation Original PowerPoint slides prepared by S. K. Mitra 3-2-65

Phase and Group Delay Example pe -For the M-point moving-average age filter the phase function is θ ( ω) = M / 2 ( M 1 ) ω + π M 2πkk μ ω 2 k= 1 M Hence its group delay is Original PowerPoint slides prepared by S. K. Mitra 3-2-66

Computing Phase and Group Delay Using MTALAB Phase delay can be computed using the function phasedelay Group delay can be computed using the function grpdelay Original PowerPoint slides prepared by S. K. Mitra 3-2-67