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PROPERTIES OF DTFT

Module by: Nguyen Huu Phuong

The discrete-time Fourier transform has many properties similar to the continuous-time Fourier transform. In the following we denote the transform pair as x(n)↔X(ω)x(n)↔X(ω) size 12{x \( n \) ↔X \( ω \) } {}.
(a) Linearity
This property states as
a 1 x 1 (n)+ a 2 x 2 (n)↔ a 1 X 1 (n)+ a 2 X 2 (n) a 1 x 1 (n)+ a 2 x 2 (n)↔ a 1 X 1 (n)+ a 2 X 2 (n) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaaWcbaGaaGymaaqabaGccaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaad6gacaGGPaGaey4kaSIaamyyamaaBaaaleaacaaIYaaabeaakiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamOBaiaacMcacqGHugYQcaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaamiwamaaBaaaleaacaaIXaaabeaakiaacIcacaWGUbGaaiykaiabgUcaRiaadggadaWgaaWcbaGaaGOmaaqabaGccaWGybWaaSbaaSqaaiaaikdaaeqaaOGaaiikaiaad6gacaGGPaaaaa@519D@ (1)
where a 1 a 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaaWcbaGaaGymaaqabaaaaa@37B2@ and a 2 a 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaaWcbaGaaGymaaqabaaaaa@37B2@ are constants. Linearity is the most fundamental property of DTFT and of many other transforms.
(b) Time reversal
x(−n)↔X(−ω) x(−n)↔X(−ω) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaeyOeI0IaamOBaiaacMcacqGHugYQcaWGybGaaiikaiabgkHiTiabeM8a3jaacMcaaaa@40F7@ (2)
(c) Time shift
x(n− n 0 )↔X(ω) e −jω n 0 x(n− n 0 )↔X(ω) e −jω n 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiabgkHiTiaad6gadaWgaaWcbaGaaGimaaqabaGccaGGPaGaeyiLHSQaamiwaiaacIcacqaHjpWDcaGGPaGaamyzamaaCaaaleqabaGaeyOeI0IaamOAaiabeM8a3jaad6gadaWgaaadbaGaaGimaaqabaaaaaaa@4887@
(d) Prequency shift
x(n) e j ω 0 n ↔X(ω− ω 0 ) x(n) e j ω 0 n ↔X(ω− ω 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiaacMcacaWGLbWaaWbaaSqabeaacaWGQbGaeqyYdC3aaSbaaWqaaiaaicdaaeqaaSGaamOBaaaakiabgsziRkaadIfacaGGOaGaeqyYdCNaeyOeI0IaeqyYdC3aaSbaaSqaaiaaicdaaeqaaOGaaiykaaaa@4889@ (3)
(e) Time domain convolution
Convolution in time domain corresponds to normal multiplication is the transform domain :
x 1 (n)* x 2 (n)↔ X 1 (ω) X 2 (ω) x 1 (n)* x 2 (n)↔ X 1 (ω) X 2 (ω) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamOBaiaacMcacaGGQaGaamiEamaaBaaaleaacaaIYaaabeaakiaacIcacaWGUbGaaiykaiabgsziRkaadIfadaWgaaWcbaGaaGymaaqabaGccaGGOaGaeqyYdCNaaiykaiaadIfadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaeqyYdCNaaiykaaaa@4ADD@ (4)
(f) Frequency domain convolution
Normal multiplication in time domain corresponds to convolution in the transform domain :
x 1 (n)* x 2 (n)↔ X 1 (ω) X 2 (ω)= 1 2π ∫ −π π X 1 ( ω , ) X 2 (ω− ω , )d ω , x 1 (n)* x 2 (n)↔ X 1 (ω) X 2 (ω)= 1 2π ∫ −π π X 1 ( ω , ) X 2 (ω− ω , )d ω , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@67C5@ (5)
(g) Parseval’s theorem
This gives the equality of energy in time domain and that in the transform domain :
∑ n=−∞ ∞ | x(n) | 2 = 1 2π ∫ −π π | X(ω) | 2 dω ∑ n=−∞ ∞ | x(n) | 2 = 1 2π ∫ −π π | X(ω) | 2 dω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaamaaqahabaWaaqWaaeaacaWG4bGaaiikaiaad6gacaGGPaaacaGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaaqaaiaad6gacqGH9aqpcqGHsislcqGHEisPaeaacqGHEisPa0GaeyyeIuoakiabg2da9maalaaabaGaaGymaaqaaiaaikdacqaHapaCaaWaa8qmaeaadaabdaqaaiaadIfacaGGOaGaeqyYdCNaaiykaaGaay5bSlaawIa7amaaCaaaleqabaGaaGOmaaaaaeaacqGHsislcqaHapaCaeaacqaHapaCa0Gaey4kIipakiaadsgacqaHjpWDaaa@5ADF@ (6)
The LHS is the energy of the signal where ∣x(n)∣2∣x(n)∣2 size 12{ lline x \( n \) rline rSup { size 8{2} } } {} is the mormalized power (power per unit resistance) of the signal. If x(n) is real we can write x2(n)x2(n) size 12{x rSup { size 8{2} } \( n \) } {} instead of ∣x(n)∣2∣x(n)∣2 size 12{ lline x \( n \) rline rSup { size 8{2} } } {}.
Figure 1: Main properties of the DTFT where x(n)↔X(ω)x(n)↔X(ω) size 12{x \( n \) ↔X \( ω \) } {} is the transform pair
The RHS is the energy of the signal based on its frequency spectrum, where ∣X(ω)∣2∣X(ω)∣2 size 12{ lline X \( ω \) rline rSup { size 8{2} } } {} is the power spectral density (PSD) (power per unit frequency), and ∣X(ω)∣2/2π∣X(ω)∣2/2π size 12{ lline X \( ω \) rline rSup { size 8{2} } /2π} {} is the energy spectral density (ESD). Figure 1 summarizes the main properties.
Example 1 
Find the DTFT of the symmetric decaying exponential x(n)= a | n |   −1<a<1 x(n)= a | n |   −1<a<1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiaacMcacqGH9aqpcaWGHbWaaWbaaSqabeaadaabdaqaaiaad6gaaiaawEa7caGLiWoaaaGccaaMf8UaaGzbVlabgkHiTiaaigdacqGH8aapcaWGHbGaeyipaWJaaGymaaaa@47D3@
Solution
This is an example about the linearity property. We separate the signal into two coponents
x(n)= x 1 (n)+ x 2 (n) x(n)= x 1 (n)+ x 2 (n) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiaacMcacqGH9aqpcaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaad6gacaGGPaGaey4kaSIaamiEamaaBaaaleaacaaIYaaabeaakiaacIcacaWGUbGaaiykaaaa@438B@
where one component is a function of exponent n and the other of –n:
x 1 (n)= a n  , n≥0    0 ,  n<0 x 1 (n)= a n  , n≥0    0 ,  n<0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamiEamaaBaaaleaacaaIXaaabeaakiaacIcacaWGUbGaaiykaiabg2da9iaadggadaahaaWcbeqaaiaad6gaaaGccaaMf8UaaiilaiaaywW7caWGUbGaeyyzImRaaGimaaqaaaqaaiaaywW7caaMf8UaaGzbVlaaicdacaaMf8UaaiilaiaaywW7caaMe8UaamOBaiabgYda8iaaicdaaaaa@51EA@
x 2 (n)= a −n  , n<0    0 ,  n≥0 x 2 (n)= a −n  , n<0    0 ,  n≥0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamiEamaaBaaaleaacaaIYaaabeaakiaacIcacaWGUbGaaiykaiabg2da9iaadggadaahaaWcbeqaaiabgkHiTiaad6gaaaGccaaMf8UaaiilaiaaywW7caWGUbGaeyipaWJaaGimaaqaaaqaaiaaywW7caaMf8UaaGzbVlaaicdacaaMf8UaaiilaiaaywW7caaMe8UaamOBaiabgwMiZkaaicdaaaaa@52D8@
Now
X 1 (ω)= ∑ n=0 ∞ a n e −jωn = ∑ n=0 ∞ (a e −jω ) n X 1 (ω)= ∑ n=0 ∞ a n e −jωn = ∑ n=0 ∞ (a e −jω ) n MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfadaWgaaWcbaGaaGymaaqabaGccaGGOaGaeqyYdCNaaiykaiabg2da9maaqahabaGaamyyamaaCaaaleqabaGaamOBaaaakiaadwgadaahaaWcbeqaaiabgkHiTiaadQgacqaHjpWDcaWGUbaaaaqaaiaad6gacqGH9aqpcaaIWaaabaGaeyOhIukaniabggHiLdGccqGH9aqpdaaeWbqaaiaacIcacaWGHbGaamyzamaaCaaaleqabaGaeyOeI0IaamOAaiabeM8a3baakiaacMcadaahaaWcbeqaaiaad6gaaaaabaGaamOBaiabg2da9iaaicdaaeaacqGHEisPa0GaeyyeIuoaaaa@5991@
Apply the formula of infinite geometric series to get
X 1 (ω)= 1 1−a e −jω  ,  | a e −jω |<1 X 1 (ω)= 1 1−a e −jω  ,  | a e −jω |<1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfadaWgaaWcbaGaaGymaaqabaGccaGGOaGaeqyYdCNaaiykaiabg2da9maalaaabaGaaGymaaqaaiaaigdacqGHsislcaWGHbGaamyzamaaCaaaleqabaGaeyOeI0IaamOAaiabeM8a3baaaaGccaaMf8UaaiilaiaaywW7caaMf8+aaqWaaeaacaWGHbGaamyzamaaCaaaleqabaGaeyOeI0IaamOAaiabeM8a3baaaOGaay5bSlaawIa7aiabgYda8iaaigdaaaa@53ED@
The convergent condition ∣ae−jω∣<1∣ae−jω∣<1 size 12{ lline ital "ae" rSup { size 8{ - jω} } rline <1} {} means ∣a∣<1∣a∣<1 size 12{ lline a rline <1} {} which is satisfied.
Similarly,
X 2 (ω)= ∑ n=−∞ −1 x 2 (n) e −jωn = ∑ n=−∞ −1 (a e jω ) −n = ∑ n=1 ∞ (a e jω ) n X 2 (ω)= ∑ n=−∞ −1 x 2 (n) e −jωn = ∑ n=−∞ −1 (a e jω ) −n = ∑ n=1 ∞ (a e jω ) n MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@6E1B@
Or
X 2 (ω)=a e jω [ ∑ n=0 ∞ (a e jω ) n ] X 2 (ω)=a e jω [ ∑ n=0 ∞ (a e jω ) n ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaeqyYdCNaaiykaiabg2da9iaadggacaWGLbWaaWbaaSqabeaacaWGQbGaeqyYdChaaOWaamWaaeaadaaeWbqaaiaacIcacaWGHbGaamyzamaaCaaaleqabaGaamOAaiabeM8a3baakiaacMcadaahaaWcbeqaaiaad6gaaaaabaGaamOBaiabg2da9iaaicdaaeaacqGHEisPa0GaeyyeIuoaaOGaay5waiaaw2faaaaa@5036@
resulting in
X 2 (ω)= a e jω 1−a e jω  ,  | a e jω |<1 X 2 (ω)= a e jω 1−a e jω  ,  | a e jω |<1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaeqyYdCNaaiykaiabg2da9maalaaabaGaamyyaiaadwgadaahaaWcbeqaaiaadQgacqaHjpWDaaaakeaacaaIXaGaeyOeI0IaamyyaiaadwgadaahaaWcbeqaaiaadQgacqaHjpWDaaaaaOGaaGzbVlaacYcacaaMf8UaaGzbVpaaemaabaGaamyyaiaadwgadaahaaWcbeqaaiaadQgacqaHjpWDaaaakiaawEa7caGLiWoacqGH8aapcaaIXaaaaa@561C@
The convergent condition is satisfied as above.
On combining the two results we obtain the final transform
X(ω)= X 1 (ω)+ X 2 (ω)= 1− a 2 1−2acos⁡ω+ a 2 X(ω)= X 1 (ω)+ X 2 (ω)= 1− a 2 1−2acos⁡ω+ a 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfacaGGOaGaeqyYdCNaaiykaiabg2da9iaadIfadaWgaaWcbaGaaGymaaqabaGccaGGOaGaeqyYdCNaaiykaiabgUcaRiaadIfadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaeqyYdCNaaiykaiabg2da9maalaaabaGaaGymaiabgkHiTiaadggadaahaaWcbeqaaiaaikdaaaaakeaacaaIXaGaeyOeI0IaaGOmaiaadggaciGGJbGaai4BaiaacohacqaHjpWDcqGHRaWkcaWGHbWaaWbaaSqabeaacaaIYaaaaaaaaaa@54EB@
It is interesting to notice that for 0 < a < 1 the magnitude spectrum concentrates at low frequencies, whereas for -1 < a < 0 the magnitude spectrum concentrates at high frequencies.
Example 2 
Shift the digital rectangular pulse of Example towards the future to become causal (Figure 2a). Now the signal has 2N or M samples. Find its DTFT transform.
Solution
The causal signal is
x(n)=A ,  0≤n≤2N=M    0 ,  otherwise x(n)=A ,  0≤n≤2N=M    0 ,  otherwise MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamiEaiaacIcacaWGUbGaaiykaiabg2da9iaadgeacaaMf8UaaiilaiaaywW7caaMf8UaaGimaiabgsMiJkaad6gacqGHKjYOcaaIYaGaamOtaiabg2da9iaad2eaaeaaaeaacaaMf8UaaGzbVlaaywW7caaIWaGaaGzbVlaacYcacaaMf8UaaGzbVlaad+gacaWG0bGaamiAaiaadwgacaWGYbGaam4DaiaadMgacaWGZbGaamyzaaaaaa@5C1F@
The total number of samples is 2N + 1 (or M + 1).
Figure 2: Example (the shifted rectangular pulse and spectra)
Applying the time shift property we obtain the transform
X c (ω)=X(ω) e −jωN =A[ 1+2 ∑ n=1 N cos⁡nω ] e −jNω X c (ω)=X(ω) e −jωN =A[ 1+2 ∑ n=1 N cos⁡nω ] e −jNω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfadaWgaaWcbaGaam4yaaqabaGccaGGOaGaeqyYdCNaaiykaiabg2da9iaadIfacaGGOaGaeqyYdCNaaiykaiaadwgadaahaaWcbeqaaiabgkHiTiaadQgacqaHjpWDcaWGobaaaOGaeyypa0JaamyqamaadmaabaGaaGymaiabgUcaRiaaikdadaaeWbqaaiGacogacaGGVbGaai4Caiaad6gacqaHjpWDaSqaaiaad6gacqGH9aqpcaaIXaaabaGaamOtaaqdcqGHris5aaGccaGLBbGaayzxaaGaamyzamaaCaaaleqabaGaeyOeI0IaamOAaiaad6eacqaHjpWDaaaaaa@5CBC@ (7)
With A = 1 and N = 2, then
X c (ω)=[ 1+2cos⁡ω+2cos⁡2ω ] e −j2ω X c (ω)=[ 1+2cos⁡ω+2cos⁡2ω ] e −j2ω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfadaWgaaWcbaGaam4yaaqabaGccaGGOaGaeqyYdCNaaiykaiabg2da9maadmaabaGaaGymaiabgUcaRiaaikdaciGGJbGaai4BaiaacohacqaHjpWDcqGHRaWkcaaIYaGaci4yaiaac+gacaGGZbGaaGOmaiabeM8a3bGaay5waiaaw2faaiaadwgadaahaaWcbeqaaiabgkHiTiaadQgacaaIYaGaeqyYdChaaaaa@516D@
The amplitude spectrum is the same as before (Figure (c)). As for the phase spectrum it is understood that if the function in the brackets is possitive then the phase is Φ(ω)=−2ωΦ(ω)=−2ω size 12{Φ \( ω \) = - 2ω} {}, whereas if the function is negative then the phase is Φ(ω)=−2ω+πΦ(ω)=−2ω+π size 12{Φ \( ω \) = - 2ω+π} {} (Figure 2c)(more in section ).
On the other hand we can take the direct transform (without using the time shift property):
X(ω)= ∑ n=0 2N A e −jωn X(ω)= ∑ n=0 2N A e −jωn MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIfacaGGOaGaeqyYdCNaaiykaiabg2da9maaqahabaGaamyqaiaadwgadaahaaWcbeqaaiabgkHiTiaadQgacqaHjpWDcaWGUbaaaaqaaiaad6gacqGH9aqpcaaIWaaabaGaaGOmaiaad6eaa0GaeyyeIuoaaaa@47E0@
Using the formula of finitive geometric series to get
X(ω)=A 1− e −j(2N+1)ω 1− e −jω     =A sin⁡ 2N+1 2 ω sin⁡ω/2 e −jNω  ,  ω≠0     =AM ,         ω=0 X(ω)=A 1− e −j(2N+1)ω 1− e −jω     =A sin⁡ 2N+1 2 ω sin⁡ω/2 e −jNω  ,  ω≠0     =AM ,         ω=0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@8FD8@ (8)

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