Chapter 3 Chapter 3 Two major topics of this chapter: Chapter 3A ●●● Discrete-Time Fourier Transform ●●垂 ●●● ●●●● Discrete-Time Frequency Response of an LTI Discrete- Discrete-Time Time Systems(DTFT of the impulse Fourier Transform(DTFT) Fourier Transform ● response) Part A:DTFT 1.1 Definition of CTFT 1.1 Definition of CTFT 1.The Continuous-Time Fourier Transform Definition Definition 1.1 Definition 1.2 Energy Density Spectrum The inverse CTFT of a Fourier Transform 1.3 Band-limited Continuous-Time Signals The CTFT of a continuous-time signal x (r)is X(js)is given by 2.The Discrete-Time Fourier Transform given by 2.1 Definition 2.2 Convergence Condition x.()-(em 无0=上xUm严am 2.3 DFT Properties 2.4 Energy Density Spectrum 2.5 DTFT Computation Using MATLAB Often referred to as the Fourier Spectrum or Often referred to as the Fourier integral 2.6 Linear Convolution Using DTFT simply the Spectrum of the continuous-time .A CTFT pair will be denoted as 27 DTFT for Special Sequence signal x(t)cmrX(j) 4
Chapter 3 Chapter 3 Discrete-Time Fourier Transform (DTFT) Chapter 3 Two major topics of this chapter: wo major topics o f this c hapter: Discrete Discrete-Time Fourier Transform Time Fourier Transform Frequency Response of an LTI Discrete Frequency Response of an LTI DiscreteTime Systems Time Systems (DTFT of the impulse DTFT of the impulse impulse response) 2 Chapter 3A Chapter 3A Discrete-Time Fourier Transform Part A: DTFT 1. The Continuous The Continuous-Time Fourier Transform Time Fourier Transform 1.1 Definition 1.2 Energy Density Spectrum 1.3 Band-limited Continuous limited Continuous-Time Signals Time Signals 2. The Discrete The Discrete-Time Fourier Transform Time Fourier Transform 2.1 Definition 2.2 Convergence Condition 2.3 DFT Properties 2.4 Energy Density Spectrum 2.5 DTFT Computation Using MATLAB 2.6 Linear Convolution Using DTFT 2.7 DTFT for Special Sequence 4 1.1 Definition of CTFT Definition The CTFT of a continuous-time signal xa(t) is given by given by ( ) () j t Xa a j x t e dt Often referred to as the Fourier Spectrum or simply the Spectrum of the continuous time simply the Spectrum of the continuous-time signal 5 1.1 Definition of CTFT Definition Th i CTFT f F i T f The inverse CTFT of a Fourier Transform Xa(j) is given by 1 () ( ) 2 j t a a xt X j e d Often referred to as the Fourier integral A CTFT i ill b d d 2 A CTFT pair will be denoted as xt X j () ( ) CTFT 6 a a xt X j
1.1 Definition of CTFT 1.1 Definition of CTFT 1.1 Definition of CTFT ·Ωis real and denotes the continuous-time .The quantity (j)is called the magnitude angular frequency variable in radians Dirichlet Conditions spectrum and the quantity (is called the In general,the CTFT is a complex function of (a)The signal x (t)has a finite number of phase spectrum in the range-oo<n<oo discontinuities and a finite number of maxima Both spectrums are real functions of and minima in any finite interval It can be expressed in the polar form as X.(j)=X( .In general,the CTFT X(j)exists if x (t) (b)The signal is absolutely integrable,i.e. satisfies the Dirichlet Conditions(狄利克雷条 where 8.(2)=ag{X.(j2} given on the next slide koa<西 1.1 Definition of CTFT 1.2 Energy Density Spectrum 1.2 Energy Density Spectrum If the Dirichlet Conditions are satisfied,then The total energy &of a finite energy .Interchanging the order of the integration,we 1x(O)edo continuous-time complex signal x(t)is given get 2元e by converges to x(r)at values of t except at ￡=k.mfh=x0x0d .-emdjdn values oft where x(f)has discontinuities The above expression can be rewritten as It can be shown that ifx(r)is absolutely 上CU.URMn integrable,then (j)<o proving the existence of the CTFT ￡-o2上m -a比0mram
1.1 Definition of CTFT is real and denotes the continuous-time angul f i bl i di lar frequency variable in radians In g p eneral, the CTFT is a complex function of in the range << It can be expressed in the It can be expressed in the polar form polar form as ( ) () () a j Xa a j X j e where () () a a j j ( ) arg{ ( )} a a X j 7 1.1 Definition of CTFT The quantity |Xa(j)| is called the magnitude a spectrum and the quantity a() is called the phase spectrum phase spectrum Both spectrums are real functions of In general, the CTFT Xa(j) exists if xa(t) satisfies the satisfies the Dirichlet Dirichlet Conditions Conditions (⣴ݻ࡙䴧ᶑ Ԧ) given on the next slide 8 1.1 Definition of CTFT Dirichlet Conditions (a) The signal xa(t) has a finite number of di ti iti d fi it b f i discontinuities and a finite number of maxima and minima in any finite interval (b) The signal is absolutely integrable, i.e. ( ) a x t dt 9 1.1 Definition of CTFT If the Dirichlet Conditions are satisfied, then 1 ( ) 2 j t Xa j ed converges to xa(t) at values of t except at l f t h (t) h di ti iti 2 values of t where xa(t) has discontinuities It can be shown that if xa(t) is absolutely integrable, then |Xa(j)| < proving the existence of the CTFT 10 existence of the CTFT 1.2 Energy Density Spectrum The total energy x of a finite energy continuous-time complex signal xa(t) is given by 2 * () () () x a aa x t dt x t x t dt The above expression can be rewritten as 1 * () ( ) 2 j t xa a x t X j e d dt 11 1.2 Energy Density Spectrum Interchangg g in the order of the integration, we get 1 * j t 1 * ( ) () 2 1 j x t X j x t e dt d a a 1 * ( ) 2 ( ) Xj Xjd a a 1 2 ( ) 2 a X d j 12 2
1.3 Band-limited Continuous-Time 1.2 Energy Density Spectrum 1.2 Energy Density Spectrum Signals 。Hence .The quantity ()Pis called the energy density spectrum ofx()and usually denoted A full-band,finite-energy,continuous-time signal has a spectrum occupying the whole kF-o as S.(2)=K.j2f frequency range-oo<<oo A band-limited continuous-time signal has a The above relation is more commonly known .The energy over a specified range of as the Parseval's relation for finite energy spectrum that is limited to a portion of the frequencies <can be computed using frequency range<<o continuous-time signals ⊥s.d 13 14 1.3 Band-limited Continuous-Time 1.3 Band-limited Continuous-Time 1.3 Band-limited Continuous-Time Signals Signals Signals An ideal band-limited signal has a spectrum Band-limited signals are classified according .A highpass,continuous-time signal has a that is zero outside a finite frequency range to the frequency range where most of the spectrum occupying the frequency range 0< gos.that is signal's is concentrated so where the bandwidth of the signal is .A lowpass,continuous-time signal has a from to co X(j2)= [0,0≤l≤2. l0,2.≤2≤∞ spectrum occupying the frequency range A bandpass,continuous-time signal has a owhere is called the bandwidth spectrum occupying the frequency range 0< However,an ideal band-limited signal cannot of the signal sO where u is the bandwidth be generated in practice
1.2 Energy Density Spectrum Hence 2 2 1 x t dt X j d () ( ) Th b l ti i l k () ( ) 2 a a x t dt X j d The above relation is more commonly known as the Parseval’s Parseval’s relation relation for finite energy continuous-time signals 13 1.2 Energy Density Spectrum The quantity |Xa(j)|2 is called the energy d it t ensity spectrum of xa(t) and ll d t d d usually denoted as 2 Th ifi d f () ( ) xx a S Xj The energy over a specified range of frequencies ab can be computed using 1 ( ) 2 b x r xx S d 14 , ( ) 2 a x r xx 1.3 Band-limited Continuous-Time Signals Afull-band, finite-energy, continuous ntinuous-time signal has a spectrum occupying the whole frequency range frequency range << A band-limited limited continuous-time signal has a spectrum that is limited to a portion of the frequency range frequency range << 15 1.3 Band-limited Continuous-Time Signals Anideal band-limited signal has a spectrum that is zero outside a finite frequency range || that is a||b, that is 0, 0 ( ) a X j ( ) 0, a b X j However, an ideal band-limited signal cannot be generated in practice 16 g p 1.3 Band-limited Continuous-Time Signals Band-limited sig g nals are classified according to the frequency range where most of the signal s’ is concentrated is concentrated A lowpass, continuous-time signal has a spectrum occupying the frequency range ||p< where p | | p p is called the bandwidth of the signal 17 1.3 Band-limited Continuous-Time Signals A hig ph ass, continuous-time signal has a spectrum occupying the frequency range 0<p ||< where the where the bandwidth bandwidth of the signal is of the signal is from p to A bandpass, continuous-time signal has a sp py g q y g ectrum occupying the frequency range 0<L || H < where HL is the bandwidth 18
2.1 Definition of DTFT 2.1 Definition of DTFT 2.1 Definition of DTFT Definition .X()|is called the magnitude function and For a real sequence x(n).X(ei)and The discrete-time Fourier transform(DTFT) @)is called the phase function Re[X(el)]are even functions of @whereas, X(ef)of a sequence x(n)is given by .In many applications,the DTFT is called the (@and Im[X(eio)]are odd functions of X(e)=>x(n)e- Fourier spectrum Note that,for any integerk In general,X(ef)is a complex function of the Likewise,X(ei)and @are called the magnitude and phase spectra X(e)=Y(ei)eja) real variable o and can be written as X(ei)=X(e)e It should be noted that DTFT is a continuous =X(e)eAa) function ofo 19 20 2.1 Definition of DTFT 2.1 Definition of DTFT 2.1 Definition of DTFT The phase function @)cannot be uniquely Example Simulation Results specified for any DTFT The magnitude and phase of the DTFT The DTFT of the unit sample sequence fo(n) X(ef)=1/(1-0.5e-j)are shown below Unless otherwise stated,we shall assume that is given by the phase function is restricted to the X(e)=>8(n)e-=5(0)=1 following range of values: Consider the causal sequence x(n)=a"u(n) -π≤o)≤元 a水1 called the principal value Xe=2==,1 =0 1-ae▣ 名 23
2.1 Definition of DTFT Definition The discrete discrete-time Fourier transform time Fourier transform (DTFT) X(ej) of a sequence of a sequence x(n) is given by is given by ( ) () j j n X e xne In general, X(ej) is a complex function of the real variable and can be written as n real variable and can be written as ( ) () () j jj Xe Xe e 19 2.1 Definition of DTFT | X(ej) | is called the magnitude function and () is call d th lled the ph f ti ase function In many pp , a lications, the DTFT is called the Fourier spectrum Lik i | X( j Likewise, | X(e ) | d ( ) ll d th j) | and () are called the magnitude and phase spectra It should be noted that DTFT is a continuous function of 20 function of 2.1 Definition of DTFT For a real sequence x(n), | X(ej) | and Re[X(ej)] are even functions of , whereas, () and Im[X(ej ( ) [ ( )] are odd functions of Note that, for any integer k ( ) () () j jj Xe Xe e ( )2 ( ) j j k Xe e 21 2.1 Definition of DTFT The phase function () cannot be uniq y uel specified for any DTFT Unless otherwise stated we shall assume that Unless otherwise stated, we shall assume that the phase function () is restricted to the following range of values: following range of values: ( ) called the principal value ( ) 22 2.1 Definition of DTFT Example The DTFT of the unit sample sequence {(n)} is given by is given by ( ) ( ) (0) 1 j jn n X e ne Consider the causal sequence x(n)=anu(n) |a|<1 n 1 ( ) j n jn X 0 1 ( ) 1 j n jn j n X e ae ae 23 2.1 Definition of DTFT Simulation Results The magnitude and phase of the DTFT The magnitude and phase of the DTFT X(ej)=1/(1ˉ0.5ej) are shown below Phase Response 0.5 Phase Response s 2 Magnitude Response 0 e in Radian s 1.5 agnitude -0 5 Phas e 1 M a 24 -3 -2 -1 0 1 2 3 -0.5 Normalized Frequency -3 -2 -1 0 1 2 3 0.5 Normalized Frequency
2.1 Definition of DTFT 2.2 Convergence Condition 2.3 DTFT Properties ●Linearity The DTFT of a sequence x(n),is a continuous If x(n)is an absolutely summable sequence, function of @It is also a periodic function of i.e.,if with a period 2 m< Shifting (in time and in frequency domain) ●Differentiation Then r(e） .The Inverse discrete-time Fourier transform x(n)台j e一=区e2ok do (IDTFT)of X(e)is given by Convolution(in time and in frequency domain) r(n)= 1X(e")edo Thus,the absolute summability ofx(n)is a xm)*m)台x(ey(e)月 Proof sufficient condition for the existence of the DTFT no台，xe)Y(em）a0 2.3 DTFT Properties 2.3 DTFT Properties 2.3 DTFT Properties Some Coemmon Discrete-Time Fourler Transform Pairs Area Theorem(simple but useful) Symmetry Relations(DTFT pairs) Tratedorm 1 0=X(e)do x(e-立m x(m台X_(em) ix,(n)台X(em) x.(m)台X(e) x(n)台X,(e) .2w+2k ●Parseval's Theorem 六+，+2 For an arbitrary real sequence 立m=2 [x(r()a@ X(ei)=X'(e-) ∑22-+2= Corollary-Energy is preserved Corollary ,a-十2ak十t十m十2ak] 1,0≤m≤ 2m2云.x(-Ydo X.(e)X,(e)X(e)arg[x(e)]
2.1 Definition of DTFT The DTFT of a sequence The DTFT of a sequence x(n), is a continuous continuous function of . It is also a periodic function of with a period with a period 2 The Inverse discrete Inverse discrete-time Fourier transform time Fourier transform (IDTFT) f X( j (IDTFT) of X(e ) ii b j) is given by 1 () ( ) j jn X d Proof - () ( ) 2 j jn x n X e e d 25 2.2 Convergence Condition If x(n) is an absolutely summable sequence, i if .e., if ( ) n x n Then n ( ) () () j jn X Thus the absolute summability of x(n) is a - - ( ) () () j jn n n X e xne xn Thus, the absolute summability of x(n) is a sufficient condition for the existence of the DTFT 26 2.3 DTFT Properties Linearity Shifting (i ti d i f d i ) (in time and in frequency domain) Differentiation j dX e ( ) j dX e nx n j d Convolution (in time and in frequency domain) () () j j xn yn X e Y e xn yn X e Y e () () 1 ( -) ()() j j xnyn X e Y e d 27 ()() 2 xnyn X e Y e d 2.3 DTFT Properties Area Theorem (simple but useful) 1 0 - ( ) j n X e xn - 1 (0) 2 j x Xe d Parseval’s Parseval’s Theorem Theorem * * 1 j j * * - - 1 () () 2 j j n x n y n Xe Y e d Corollary——Energy is preserved 2 2 1 ( ) j X d 28 2 ( ) 2 j n x n X e d 2.3 DTFT Properties Symmetry Relations (DTFT pairs) () ( ) j r cs xn X e () ( ) j i ca jx n X e () ( ) j R xn Xe () ( ) j I x n jX e For an arbitrary real sequence () ( ) cs R xn Xe () ( ) ca I x n jX e Corollary * () ( ) j j Xe X e Corollary ( ) j XR e ( ) j X e ( ) j XI e arg ( ) j X e R 29 2.3 DTFT Properties 30
2.3 DTFT Properties 2.3 DTFT Properties 2.4 Energy Density Spectrum Example Step 3:Calculate the DTFT of m(n) Determine the DFT Y(ejm)of y(n)=(n+1)a"u(n) The total energy of a finite-energy sequence ,e2-j-gje°- de-ju g(n)is given by (la<1) Step 1:Let x(n=du(n).Therefore do-aem了1-aeo 6.-lecof (nFx(m)tx(m） Step 4:Calculate the DTFT Y(ef)of y(n) Step 2:Calculate the DTFT X(ei) ae- From Parseval's relation we observe that 1 Xe）=1-e 1 (1-ae-i) 6-l(( 31 2.4 Energy Density Spectrum 2.4 Energy Density Spectrum 2.4 Energy Density Spectrum ·The quantity Recall that the autocorrelation sequence of g(n)can be expressed as As a result,the energy density spectrum S(ef) .(e）-=lc(ef 0=2gg(-1-m》=g0*g-0 of a real sequence g(m)can be computed by taking the DTFT of its autocorrelation is called the energy density spectrum .As we know that the DTFT of g(-/)is G(e-j), sequence rD).i.e.. ●The area under this curve in the range-rso≤π therefore,the DTFT of g()*g(-)is given by divided by 2 is the energy of the sequence IG(),where we have used the fact that for a s.()-E. real sequence g(n),G(e-j)=G*() 品
2.3 DTFT Properties Example Determine the DFT Y(ej) of y(n)=(n+1)anu(n) (|a| ) <1 Step 1: Let x(n)=anu(n) . Therefore y(n)=nx(n)+x(n) Step 2: Calculate the DTFT X(ej) 1 - ( ) 1 j j X e ae 31 2.3 DTFT Properties Step 3: Calculate the DTFT of nx(n) 2 2 ( ) 1 1 j jj j j dX e aje ae j j d Step 4: Calculate the DTFT Y(ej) of y(n) 1 1 d j j ae ae p ( ) y( ) 2 2 1 1 1 j j j j j ae Y e 2 2 1 1 1 j j j ae ae ae 32 2.4 Energy Density Spectrum The total energy of a finite-energy sequence g(n) is given by 2 ( ) F P l’ l ti b th t g n g n From Parseval’s relation we observe that 2 2 1 ( ) j G d ( ) 2 j g n g n G e d 33 2.4 Energy Density Spectrum The quantity The quantity 2 j j gg S e Ge is called the energy density spectrum gg The area under this curve in the range di id d b i th f th divided by 2 is the energy of the sequence 34 2.4 Energy Density Spectrum Recall that the autocorrelation sequence rgg(l) of g(n) can be expressed as r l gng l n gl g l ( ) ( ) ( ( )) ( ) ( ) As we know that the DTFT of g(ˉl) is G(eˉj) ( ) ( ) ( ( )) ( ) ( ) gg n r l gng l n gl g l As we know that the DTFT of g( l) is G(e j ), therefore, the DTFT of is given by |G( j)|2 here e ha e sed the fact that for a g() ( ) l g l |G(ej)|2, where we have used the fact that for a real sequence g(n), G(eˉj)=G*(ej) 35 2.4 Energy Density Spectrum A lt th d it t S ( j As a result, the energy density spectrum S ) gg(ej) of a real sequence g(n) can be computed by t ki th DTFT f it t l ti taking the DTFT of its autocorrelation sequence rgg(l), i.e., ( ) j j l gg gg l S e r le l 36
2.4 Energy Density Spectrum 2.4 Energy Density Spectrum 2.5 DTFT Computation Using MATLAB Example Compute the energy of the sequence .Therefore,Compute the energy of the The function freqz can be used to compute he(n)=sinon the values of the DTFT of a sequence, sequence -o≤n≤o described as a rational function in the form of Here 江 of=a(cfao 三f=六da-是< X(e）=AtBe++Pue .Hence,hn)is a finite-energy sequence do+de+dye at a prescribed set of discrete frequency points whereH(e）= 10≤asa = 0o.<回sπ 3 3 2.5 DTFT Computation Using MATLAB TT Computation Using MATLAB 2.6 Linear Convolution Using DTFT For example,the statement H=fregz(p.d.o) returns the frequency response values as a According to the convolution theorem vector H of a DTFT defined in terms of the vectors p and d containing the coefficientsp m)=x(m)*hm)÷Y(em)=X(e)H(em） and,respectively at a prescribed set of frequencies between 0 and 2n given by the An implication of this result is that the linear vector o. convolution wn)of the sequencesx(n)and h()can be performed as follows: For example p=[0.008-0.0330.05-0.0330.008] =[12.372.71.60.41] 40 42
2.4 Energy Density Spectrum Example Compute the energy of the sequence Compute the energy of the sequence sin ( ) cn hn n Here ( ) , LP hn n n 2 2 1 ( ) j h d 2 - 1 ( ) 2 j LP LP n h n He d where 1, 0 0 j c H e LP 37 0, c 2.4 Energy Density Spectrum Therefore Compute the energy of the Therefore, Compute the energy of the sequence 2 - 1 ( ) 2 cc c LP hn d Hence, hLP(n) is a finite-energy sequence energy sequence 2 c n 38 2.5 DTFT Computation Using MATLAB 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 - - 0 1 j jM j M j jN p pe p e X e at a prescribed set of discrete frequency points - - 0 1 j jN N X e d de d e at a prescribed set of discrete frequency points = l . 39 2.5 DTFT Computation Using MATLAB For example, the statement H= freqz(p,d,) returns the frequency response values as a returns the frequency response values as a vector H of a DTFT defined in terms of the vectors vectors p and d containing the coefficients containing the coefficients {p }i and {di} , respectively at a prescribed set of frequencies between frequencies between 0 and 2 given by the given by the vector . For example p=[0.008 ˉ0.033 0.05 ˉ0.033 0.008] 40 d=[1 2.37 2.7 1.6 0.41] 2.5 DTFT Computation Using MATLAB Rl t Ii t 0.5 1 Real part d e0.5 1 Imaginary part d e -0.5 0 Amplitu d -0.5 0 Amplitu d 0 0.2 0.4 0.6 0.8 1 -1 / 0 0.2 0.4 0.6 0.8 1 -1 / Magnitude Spectrum Phase Spectrum 0.6 0.8 1 itude 0 2 4 radians 0 0.2 0.4 Magn 4 -2 0 Phase, r 41 0 0.2 0.4 0.6 0.8 1 0 / 0 0.2 0.4 0.6 0.8 1 -4 / 2.6 Linear Convolution Using DTFT According to the convolution theorem () () () j jj yn xn hn Y e X e H e An implication of this result is that the linear p convolution y(n) of the sequences x(n) and h(n) can be performed as follows: can be performed as follows: 42
2.6 Linear Convolution Using DTFT 2.7 DTFT for Special Sequence 2.7 DTFT for Special Sequence Step 1:Compute the DTFTs X(@i)and H(ei) The DTFT can also be defined for a certain .A Dirac delta function @is a function of of the sequences x(n)and /n),respectively. class of sequences which are neither with infinite height,zero width,and unit area absolutely summable nor square summable. Step 2:Form the DTFT Y(e)=X(e)H(l) It is the limiting form of a unit area pulse Examples of such sequences are the unit step Step 3:Compute the IDTFT y(n)of Y(e) sequence u(n),the sinusoidal sequence function p(@)as A goes to zero satisfying r(n) DTFT L) cos(n+)and the exponential sequence a" IDTFT For this type of sequences,a DTFT P.oNdo=[S(oydo N DITC网 representation is possible using the Dirac delta function & 44 2.7 DTFT for Special Sequence 4 -4124/20
2.6 Linear Convolution Using DTFT Step 1: Compute the DTFTs X(ej) and H(ej Step 1: Compute the DTFTs X(e ) and H(e ) of the sequences x(n) and h(n), respectively. St 2 F th DTFT Y( j) X( j)H( j Step 2: Form the DTFT Y(e ) j)= X(ej)H(ej) Step 3: Compute the IDTFT y(n) of Y(ej p p y( ) ( ) DTFT x(n) X(ej) Y(ej) y(n) DTFT h IDTFT h(n) H(ej) ( ) y(n) 43 H(ej ) 2.7 DTFT for Special Sequence The DTFT can also be defined for a certain cl f hi h ith lass of sequences which are neither absolutely summable nor square summable. Examples of such sequences are the unit step sequence u(n), q the sinusoidal sequence and the exponential sequence For this type of sequences a DTFT 0 cos( ) n n A For this type of sequences, a DTFT representation is possible using the Dirac delta function Ⱦ(ɓ) 44 function Ⱦ(ɓ) 2.7 DTFT for Special Sequence A Dirac delta function Ⱦ(ɓ) is a function of with infinite height, zero width, and unit area It is the limiting form of a It is the limiting form of a unit area pulse unit area pulse function p ( ) as goes to zero satisfying lim ( ) ( ) pd d 0 lim ( ) ( ) pd d ! 45 2.7 DTFT for Special Sequence p ( ) 1 / 2 / 2 46