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DISCRETE - TIME FOURIER SERIES (DTFS)

Module by: Nguyen Huu Phuong

The discrete-time Fourier series (DTFS) applies only for periodic signals whereas most realistic signals are aperiodic. Furthemore it does not apply to systems. These are the two reasons why the DTFS has limited use and we will go through it quickly.
A periodic signal of period N (in Figure 1 N = 8) can be expressed mathematically as
x(n)=x(n+N) , all n x(n)=x(n+N) , all n MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiaacMcacqGH9aqpcaWG4bGaaiikaiaad6gacqGHRaWkcaWGobGaaiykaiaaywW7caGGSaGaaGzbVlaadggacaWGSbGaamiBaiaaywW7caWGUbaaaa@4847@ (1)
Such a signal can be expanded into a series of N compoments:
x(n)= ∑ k=0 N−1 a k e j 2π N kn  , n=0, 1, 2,..., N−1 (synthesis equation) x(n)= ∑ k=0 N−1 a k e j 2π N kn  , n=0, 1, 2,..., N−1 (synthesis equation) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@7378@ (2)
The exponential can be written as j2πkn/Nj2πkn/N size 12{j2π ital "kn"/N} {} for convenience but written as above is more meaningful. The coefficents akak size 12{a rSub { size 8{k} } } {}are the frequency components or spectral components (coefficients) of the signal x(n). They are given by
a k = 1 N ∑ n=0 N−1 x(n) e −j 2π N kn  , k=0, 1, 2,..., N−1 (analysis equation) a k = 1 N ∑ n=0 N−1 x(n) e −j 2π N kn  , k=0, 1, 2,..., N−1 (analysis equation) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaaWcbaGaam4AaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaWGobaaamaaqahabaGaamiEaiaacIcacaWGUbGaaiykaiaadwgadaahaaWcbeqaaiabgkHiTiaadQgadaWcaaqaaiaaikdacqaHapaCaeaacaWGobaaaiaadUgacaWGUbaaaaqaaiaad6gacqGH9aqpcaaIWaaabaGaamOtaiabgkHiTiaaigdaa0GaeyyeIuoakiaaywW7caGGSaGaaGzbVlaadUgacqGH9aqpcaaIWaGaaiilaiaaysW7caaIXaGaaiilaiaaysW7caaIYaGaaiilaiaac6cacaGGUaGaaiOlaiaacYcacaaMe8UaamOtaiabgkHiTiaaigdacaaMf8UaaiikaiaadggacaWGUbGaamyyaiaadYgacaWG5bGaam4CaiaadMgacaWGZbGaaGjbVlaadwgacaWGXbGaamyDaiaadggacaWG0bGaamyAaiaad+gacaWGUbGaaiykaaaa@74F8@ (3)
The factor 1/N can be appended to the synthesis equation or to the analysis equation as above. Notice that a signal of period N sampled is expanded into the same number of spectral components. Whereas a periodic continuous-time signal is expanded into an infinite number of sinusoids. Also, the series as defined by Equation 3 is periodic with the period N, this is again quite different from the continuous-time series which is in no way periodic.
Figure 1: Period signal with period of 8 samples
To find the coefficients akak size 12{a rSub { size 8{k} } } {}we usually consider the period of the signal from n = 0 to N-1, then compute successively the real component ReakReak size 12{"Re"a rSub { size 8{k} } } {}, the imaginary compenent ImakImak size 12{"Im"a rSub { size 8{k} } } {}, the magnitude ∣ak∣∣ak∣ size 12{ lline a rSub { size 8{k} } rline } {}and the phase ΦkΦk size 12{Φ rSub { size 8{k} } } {}, where
| a k |= Re⁡ 2 ( a k )+ Im⁡ 2 ( a k ) | a k |= Re⁡ 2 ( a k )+ Im⁡ 2 ( a k ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaamyyamaaBaaaleaacaWGRbaabeaaaOGaay5bSlaawIa7aiabg2da9maakaaabaGaciOuaiaacwgadaahaaWcbeqaaiaaikdaaaGccaGGOaGaamyyamaaBaaaleaacaWGRbaabeaakiaacMcacqGHRaWkciGGjbGaaiyBamaaCaaaleqabaGaaGOmaaaakiaacIcacaWGHbWaaSbaaSqaaiaadUgaaeqaaOGaaiykaaWcbeaaaaa@4947@ (4)
Φ k =arctg Im⁡( a k ) Re⁡( a k ) Φ k =arctg Im⁡( a k ) Re⁡( a k ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfA6agnaaBaaaleaacaWGRbaabeaakiabg2da9iaadggacaWGYbGaam4yaiaadshacaWGNbWaaSaaaeaaciGGjbGaaiyBaiaacIcacaWGHbWaaSbaaSqaaiaadUgaaeqaaOGaaiykaaqaaiGackfacaGGLbGaaiikaiaadggadaWgaaWcbaGaam4AaaqabaGccaGGPaaaaaaa@4890@ (5)
Example 1 
(a) Consider a periodic sequence of 64 samples of which the first sample is the unit sample δ(t)δ(t) size 12{δ \( t \) } {} and the next 63 samples are zero (Figure 2a) Find its magnitude and phase spectrum.
(b) Now the unit sample occurs at n0n0 size 12{n rSub { size 8{0} } } {} instead at origin. Find new magnitude and phase spectrum.
Solution
(a) The spectral coefficients akak size 12{a rSub { size 8{k} } } {} are
a k = 1 N ∑ n=0 N−1 x(n) e −j 2π N kn = 1 N ∑ n=0 N−1 δ(n) e −j 2π N kn          = 1 N e −j 2π N kn | n=0 = 1 N = 1 64 a k = 1 N ∑ n=0 N−1 x(n) e −j 2π N kn = 1 N ∑ n=0 N−1 δ(n) e −j 2π N kn          = 1 N e −j 2π N kn | n=0 = 1 N = 1 64 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@8677@
Thus the magnitud spectrum equals 1/64 at every value of k (Figure 2b) and the phase spectrum equals to zero at every value of k (Figure 2c).
Figure 2: Example 1 (periodic unit sample sequence with N = 64 and its magnitude and phase spectrum)
(b) Now if the unit sample occurs at n=n0n=n0 size 12{n=n rSub { size 8{0} } } {}the spectral coefficients become
a k = 1 N ∑ n=0 N−1 δ(n− n 0 ) e −j 2π N kn = 1 N e −j 2π N k n 0 a k = 1 N ∑ n=0 N−1 δ(n− n 0 ) e −j 2π N kn = 1 N e −j 2π N k n 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaaWcbaGaam4AaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaWGobaaamaaqahabaGaeqiTdqMaaiikaiaad6gacqGHsislcaWGUbWaaSbaaSqaaiaaicdaaeqaaOGaaiykaiaadwgadaahaaWcbeqaaiabgkHiTiaadQgadaWcaaqaaiaaikdacqaHapaCaeaacaWGobaaaiaadUgacaWGUbaaaaqaaiaad6gacqGH9aqpcaaIWaaabaGaamOtaiabgkHiTiaaigdaa0GaeyyeIuoakiabg2da9maalaaabaGaaGymaaqaaiaad6eaaaGaamyzamaaCaaaleqabaGaeyOeI0IaamOAamaalaaabaGaaGOmaiabec8aWbqaaiaad6eaaaGaam4Aaiaad6gadaWgaaadbaGaaGimaaqabaaaaaaa@5CB4@
Form this the magnitude and phase spectra are
∣ a k ∣ = 1 N ∣ a k ∣ = 1 N size 12{ lline a rSub { size 8{k} } rline = { {1} over {N} } } {}
Φ k =− 2π N k n 0   radians Φ k =− 2π N k n 0   radians MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabfA6agnaaBaaaleaacaWGRbaabeaakiabg2da9iabgkHiTmaalaaabaGaaGOmaiabec8aWbqaaiaad6eaaaGaam4Aaiaad6gadaWgaaWcbaGaaGimaaqabaGccaaMf8UaaGzbVlaadkhacaWGHbGaamizaiaadMgacaWGHbGaamOBaiaadohaaaa@4A67@
Thus the magnitude spectrum is the same as before but the phase spectrum changes with k if n0n0 size 12{n rSub { size 8{0} } } {} is fixed. For example n0=1n0=1 size 12{n rSub { size 8{0} } =1} {}, the phase spectrum is
Φ k =− 2π 64 k  radians Φ k =− 2π 64 k  radians MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabfA6agnaaBaaaleaacaWGRbaabeaakiabg2da9iabgkHiTmaalaaabaGaaGOmaiabec8aWbqaaiaaiAdacaaI0aaaaiaadUgacaaMf8UaaGzbVlaadkhacaWGHbGaamizaiaadMgacaWGHbGaamOBaiaadohaaaa@492F@
The phase increases with k. At k = 0, Φ0=0Φ0=0 size 12{Φ rSub { size 8{0} } =0} {}; at k=32k=32 size 12{k="32"} {}, Φ32=−πΦ32=−π size 12{Φ rSub { size 8{"32"} } = - π} {}; at k=64k=64 size 12{k="64"} {}, Φ64=−2πΦ64=−2π size 12{Φ rSub { size 8{"64"} } = - 2π} {}. Actually the phase is understood to lie in the interval [−π,π][−π,π] size 12{ \[ - π,π \] } {}, hence at n =32 the phase reaches −π−π size 12{ - π} {} which is also ππ size 12{π} {}, afterwards the phase decreases gradually to zero instead of 2π2π size 12{2π} {}at n = 64 (Figure 3)
Figure 3: Example (the phase spectrum of the previous sequence when it is delayed by one sample)

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