Pogram P3.1 can he used to evaluale and plot the DTFT of the farm of E4 33 % Progr P3.1 Evaluation of the % compute the trequescy samples of the TFT aubplot (2,1 0 Chapter 3Discrete-Time Signals in the plot(w/ps.real0) label('Vosega /\pi'); labelplitude uplos (2.1.2 ylabel Aplitue label C'omegs Apa ylabel C'Amplitude plot (2.1.2 plot(u/ps.ngleh) label(omegs Apa labelC Phase,radie function of the MATLAB commund paae pectum 7 The june can be emoved ing the MATLAB command sorp 35 How would you modity Progran Po plat the phase in dags 3.1 Introduction In the previous two exercises you dealt with the time-domain representation of discrete-time signals and systems, and investigated their properties. Further insight into the properties of such signals and systems is obtained by their representation in the frequency-domain To this end three commonly used representations are the discrete-time Fourier transform (DTFT), the discrete Fourier transform (DFT), and the z-transform. In this exercise you will study all three representations of a discrete-time sequence. 3.2 Background Review R3.1 The discrete-time Fourier transform (DTFT) X (e) of a sequence r[n] is defined by In general X (e is a complex function of the real variable w and can be written as (3.2) where Xre(eand Xim(e) are, respectively, the real and imaginary parts of X(e) and are real functions ofw. X() can alternately be expressed in the fornm (3.3) where @w)s arg((e'")} (3.4) The quantity X(is called the magnitude function and the quantity (is called the phase function, with both functions again being real functions of w In many applications, the Fourier transform is called the Fourier spectnam and, likewise. IX(e")1 and (w) are referred to as the magnitude spectrum and phase spectrum, respectively. R3.2 The DTFT X(e) is a periodic continuous function in w with a period 2 R3.3 For a real sequence n, the real part Xre(e of its DTFT and the magnitude function X(e) are even functions of w, whereas the imaginary part Xim(e) and the phase function (w) are odd functions of w 2:07 PM Fri Mar 15 30%. r3 Discrete-Time Signals in the Frequency Domain X labmanual.pdf R3.4 The inverse discrete-time Fourier transform [n of X (e)is given by (3.5) R3.5 The Fourier transform X(e )of a sequence nexistsif[n isabsolutely summable that is, Ilm, (3.6) R3.6 The DTFT satisfies a number of useful properties that are often uitilized in a number of applications. A detailed listing of these properties and their analytical proofs can be found in any text on digital signal processing. These properties can also be verified using MATLAB. We list below a few selected properties that will be encountered later in this Time-Shifting Property If G()denotes the DTFT of a sequence in, then the DTFT of the time-shifted sequence n no is given by e (). Frequency-Shifting Property If G( denotes the DTFT of a sequence gin, then the DTFT of the sequence e"g[n is given by G(e)). Convolution Property-If G(e) and H(e denote the DTFTs of the sequences g[n] and hn.respectively, then the DTFT of the sequence gnin is given by G(e)H(). Modulation Property-I and H(denote the DTFTs of the sequences gin] and h respectively, then the DTFT of the sequence g[nhn is given by 2 Time-Reversal Property If G() denotes the DTFT of a sequence gin,then the DTFT of the time-reversed sequence g-n is given by G(e R3.7 The N-point discrete Fourier transform (DFT) of a finite-length sequence defined for 0s-1, is given by N-1 (3.7) where (3.8) R3.8 The N-point DFT Xk of a length-N sequence r[n],0,.,N-1 is simply the frequency samples of its DTFT X(e evaluated at N uniformly spaced frequency points,w2k/N,k-0,1.N 1, that is, (3.9) 3.2 Background Review 35 R3.9 The N-point circular comvolution of two length-N sequences gin and h). 0S n S N -1, is defined by (3.10) where (n)N n modulo N. The N-point circular convolution operation is usually denoted R3.10 The linear convolution of a length-N sequence g[n, 0 sn SN 1, with a length-M sequence hn, 0SnS M-1, can be obtained by a (NM-1)-point (3.12) where ge n) and he [n are obtained by appending g[n) and hin with zero-valued samples: (3.13) circular convolution of two length (N+M-1 sequences, n and hn gn. 0SnsN-1, (3.14) R3 The DFT satisfies a number of useful properties that are often utilized in a number of applications. A detailed listing of these properties and their analytical proofs can be found in any text on digital signal processing. These properties can also be verified using MATLAB. We list below a few selected properties that will be encountered later in this Cireular Time-Shifting Property-IGik] denotes the N-point DFT of a length-N sequence g[n]. then the N-point DFT of the circularly time-shifted sequence gl (n-no)is given Circular Frequency-Shifting Property If Gk] denotes the N-point DFT of a length-N sequence gm. then the N-point DFT of the sequence w k "g[n] is given by GRk-ko Circular Convolutom Property-If Glk and H k dene the N-point DFTs of the length-N sequences gn] and hn, respectively, then the N-point DFT of the circularly convolved sequence gin hin is given by GikHk Parseval's Relation-If Gk] denotes the N-point DFT of a length-N sequence gn. then N-1 N-1 (3.15) R3.12 The periodic even part gn and the periodic odd part gpo n] of a length-N real sequence g[n] are given by (3.16) 36 Chapter 3 . Discrete-Time Signals in the Frequency Domain If G[k] denotes the N-point DFofg[n], then the N-point DFTs of gpe [n] and Sp [n] are given by Re G) and ImGkrespectively. R3.13 Let g[n] and h[n] be two length-N real sequences, with GAI and HAI denoting their respective N-pont DFTs. These two N-point DFTs can be computed efficiently using a single N-point DFT Xk] of a complex length-N sequence r[n] defined by rn] gn jhn] using (3.18) (3.19) R3.14 Let n be a real sequence of length 2N with V[k] denoting its 2N-point DFT Define two real sequences g[n] and hn of length N each as g[n] = u[2n] and h[n] = v[2n + 1], 0 n M or additional (M- N) poles at0 if N M or additional (M- N) poles at0 if N