DTFT OF SOME POPULAR SIGNALS 1.3 2007/12/06 01:00:52 US/Central 2008/07/07 02:59:06.040 GMT-5 Phuong Huu Nguyen nhphuong@hcmuns.edu.vn Phuong Huu Nguyen nhphuong@hcmuns.edu.vn In this section the DTFT transform of a number of interested signals are given. (a) Unit sample δ(n) size 12{δ \( n \) } {} X(ω)=∑n=−∞∞δ(n)e−jωn=e−jω0=1 size 12{X \( ω \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {δ \( n \) e rSup { size 8{ - jωn} } } =e rSup { size 8{ - jω0} } =1} {} This is very special, the transform has real amplitude of 1 and zero phase at all frequencies. Since the transform is 2π size 12{2π} {}-periodic, all frequencies just means a period [−π,π] size 12{ \[ - π,π \] } {}.

Unit sample and its frequency spectrum

(b) Delayed unit sample δ(n−n0) size 12{δ \( n - n rSub { size 8{0} } \) } {} The transform is X(ω)=∑n=−∞∞δ(n−n0)e−jωn=e−jωn0 size 12{X \( ω \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {δ \( n - n rSub { size 8{0} } \) e rSup { size 8{ - jωn} } } =e rSup { size 8{ - jωn rSub { size 6{0} } } } } {} The magnitude and phase spectra are respectively |X(ω)|=| e −jω n 0 |=1 Φ(ω) ∠  e −jω n 0 =−ω n 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOabaeqabaGaaiiFaiaadIfacaGGOaGaeqyYdCNaaiykaiaacYhacqGH9aqpcaGG8bGaamyzamaaCaaaleqabaGaeyOeI0IaamOAaiabeM8a3jaad6gadaWgaaadbaGaaGimaaqabaaaaOGaaiiFaiabg2da9iaaigdaaeaacqqHMoGrcaGGOaGaeqyYdCNaaiykaiaaysW7cqGHGic0caaMe8UaamyzamaaCaaaleqabaGaeyOeI0IaamOAaiabeM8a3jaad6gadaWgaaadbaGaaGimaaqabaaaaOGaeyypa0JaeyOeI0IaeqyYdCNaamOBamaaBaaaleaacaaIWaaabeaaaaaa@5D0E@ The phase is proportional to frequency, but remember that the phase spectrum is understood to stay within the range [−π,π] size 12{ \[ - π,π \] } {}. (c) Decaying exponential x(n)= a n u(n),  |a| <1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiaacMcacqGH9aqpcaWGHbWaaWbaaSqabeaacaWGUbaaaOGaamyDaiaacIcacaWGUbGaaiykaiaacYcacaaMf8UaaGzbVlaacYhacaWGHbGaaiiFaiaaysW7cqGH8aapcaaIXaaaaa@49A7@ The transform is X(ω)= ∑ n=−∞ ∞ a n u(n) e −jωn = ∑ n=0 ∞ a n e −jωn MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIfacaGGOaGaeqyYdCNaaiykaiabg2da9maaqahabaGaamyyamaaCaaaleqabaGaamOBaaaakiaadwhacaGGOaGaamOBaiaacMcacaWGLbWaaWbaaSqabeaacqGHsislcaWGQbGaeqyYdCNaamOBaaaaaeaacaWGUbGaeyypa0JaeyOeI0IaeyOhIukabaGaeyOhIukaniabggHiLdGccqGH9aqpdaaeWbqaaiaadggadaahaaWcbeqaaiaad6gaaaaabaGaamOBaiabg2da9iaaicdaaeaacqGHEisPa0GaeyyeIuoakiaadwgadaahaaWcbeqaaiabgkHiTiaadQgacqaHjpWDcaWGUbaaaaaa@5D43@ Resulting in X( ω )= 1 1−a e −jω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiaadIfadaqadaqaaiabeM8a3bGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaaqaaiaaigdacqGHsislcaWGHbGaamyzamaaCaaaleqabaGaeyOeI0IaamOAaiabeM8a3baaaaaaaa@4334@ For a = 1 we have the unit step u(n). The unit step does not satisfy the existence condition Equation hence, in principle, has no DTFT. However there is this function X(ω)= 1 1− e −jω   + πδ(ω) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiaadIfacaGGOaGaeqyYdCNaaiykaiabg2da9maalaaabaGaaGymaaqaaiaaigdacqGHsislcaWGLbWaaWbaaSqabeaacqGHsislcaWGQbGaeqyYdChaaaaakiaaysW7caaMe8Uaey4kaSIaaGjbVlabec8aWjabes7aKjaacIcacqaHjpWDcaGGPaaaaa@4E39@ whose inverse transform is u(n)(d) Symmetric rectangular pulse The digital pulse (see Example ) consists of 2N + 1 samples of amplitude 1: x ( n ) = 1 , − N ≤ n ≤ N 0 , otherwise alignl { stack { size 12{x \( n \) =1 matrix { {} # {} } , matrix { {} # {} } - N <= n <= N} {} # size 12{ matrix { {} # {} # {} } 0 matrix { {} # {} } , matrix { {} # {} } ital "otherwise"} {} } } {} From Example , the transform is X(ω)=1+2∑n=1Ncosnω size 12{X \( ω \) =1+2 Sum cSub { size 8{n=1} } cSup { size 8{N} } {"cos"nω} } {} With relation (Equation) this result can be put in another the form of a radio of two simusoidal functions.

Amplitude spectrum of digital rectangular pulse with N = 5

The magnitude spectrum ansists of the main lobe having peak value of 2N + 1 and decreasing sidelobes. The origin and the zero-crossing points are separated evenly with distance of 2π/(2N+1) size 12{2π/ \( 2N+1 \) } {} ( Figure ). (e) Complex exponential and sine, cosine The complex exponential is x ( n ) = e jω 0 n − ∞ < n < ∞ size 12{x \( n \) =e rSup { size 8{jω rSub { size 6{0} } n} } matrix { {} # {} # {} } - infinity <n< infinity } {} Its DTFT transform is X(ω)=2π∑k=−∞∞δω−ω0−2πk size 12{X \( ω \) =2π Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {δ left (ω - ω rSub { size 8{0} } - 2πk right )} } {} The cosin and sine have transforms cos⁡ ω 0 n↔πδ(ω− ω 0 )+πδ(ω+ ω 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiGacogacaGGVbGaai4CaiabeM8a3naaBaaaleaacaaIWaaabeaakiaad6gacqGHugYQcqaHapaCcqaH0oazcaGGOaGaeqyYdCNaeyOeI0IaeqyYdC3aaSbaaSqaaiaaicdaaeqaaOGaaiykaiabgUcaRiabec8aWjabes7aKjaacIcacqaHjpWDcqGHRaWkcqaHjpWDdaWgaaWcbaGaaGimaaqabaGccaGGPaaaaa@538F@ sin⁡ ω 0 n↔−jπδ(ω− ω 0 )+jπδ(ω+ ω 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiGacohacaGGPbGaaiOBaiabeM8a3naaBaaaleaacaaIWaaabeaakiaad6gacqGHugYQcqGHsislcaWGQbGaeqiWdaNaeqiTdqMaaiikaiabeM8a3jabgkHiTiabeM8a3naaBaaaleaacaaIWaaabeaakiaacMcacqGHRaWkcaWGQbGaeqiWdaNaeqiTdqMaaiikaiabeM8a3jabgUcaRiabeM8a3naaBaaaleaacaaIWaaabeaakiaacMcaaaa@565F@