5.0 Discrete-time Fourier Transform Representation for discrete-time signals Chapters 3, 4, 5 Chap 3 Periodic Fourier Series Chap 4 Aperiodic Fourier Transform Chap 5 𝑥 𝑡 ↔𝑋 𝑗𝜔 𝑥[𝑛]↔𝑋( 𝑒 𝑗𝜔 ) Continuous 𝑥 𝑡 =𝑥(𝑡+𝑇) Discrete 𝑥[𝑛]=𝑥[𝑛+𝑁]

Fourier Transform (p.3 of 4.0) FS 𝜔 𝑡 𝜔 0 3𝜔 0 𝜔 0 = 2𝜋 𝑇 𝑒 𝑗3 𝜔 0 𝑡 2𝜔 0 T→∞ 𝜔 0 →0 periodic in 𝑡 discrete in 𝜔 𝑥(𝑡) 𝑋(𝑗𝜔) 𝐹 𝑡 𝑗𝜔 𝜔 𝑘 𝑒 𝑗 𝜔 𝑘 𝑡 aperiodic in 𝑡 continuous in 𝜔

Discrete-time Fourier Transform 𝑁 (2𝜋) 𝑁 N dim N variables 𝐹𝑆 𝑘 𝑛 012 𝜔 0 periodic in 𝑛 𝜔 2𝜔 0 𝜔 0 𝜔 0 = 2𝜋 𝑁 ∞ dim 𝜔 0 →0 𝑁→∞ discrete and periodic in 𝜔 ∞ variables 𝑥[𝑛] 2𝜋 𝑋( 𝑒 𝑗𝜔 ) 𝐼𝑚 𝐹 𝜔 𝑛 𝑅𝑒 𝜔 𝑘 𝑒 𝑗𝜔 (1,0) 𝜔 𝑘 +2𝜋 aperiodic in 𝑛 𝑒 𝑗 𝜔 𝑘 𝑛 =𝑒 𝑗( 𝜔 𝑘 +2𝜋)𝑛 continuous and periodic in 𝜔

Harmonically Related Exponentials for Periodic Signals (p.11 of 3.0) 𝑉={𝑥(𝑡) 𝑥(𝑡) periodic, fundamental period [𝑛] =𝑇(𝑁)} T (𝑁) 𝑡, 𝑛 𝑎 4 𝑡, 𝑛 𝑎 1 𝑎 3 𝑎 5 𝑎 0 𝑘=1 𝑘 𝜔 0 = 2𝜋 𝑇 (𝑁) 0 1 2 3 4 5 𝑡, 𝑛 𝜔 𝑘=2 𝜔 0 3𝜔 0 5𝜔 0 𝑡, 𝑛 2𝜔 0 4𝜔 0 𝑘=3 All with period T: integer multiples of ω0 Discrete in frequency domain (𝑁)

From Periodic to Aperiodic Considering x[n], x[n]=0 for n > N2 or n < -N1 Construct

From Periodic to Aperiodic Considering x[n], x[n]=0 for n > N2 or n < -N1 Fourier series for Defining envelope of

As signal, time domain, Inverse Discrete-time Fourier Transform spectrum, frequency domain Discrete-time Fourier Transform Similar format to all Fourier analysis representations previously discussed

Fourier Transform pair, different expressions (p.10 of 4.0) spectrum, frequency domain Fourier Transform signal, time domain Inverse Fourier Transform Fourier Transform pair, different expressions very similar format to Fourier Series for periodic signals

Note: X(ejω) is continuous and periodic with period 2 Integration over 2 only Frequency domain spectrum is continuous and periodic, while time domain signal is discrete-time and aperiodic Frequencies around ω=0 or 2 are low-frequencies, while those around ω=  are high-frequencies, etc. See Fig. 5.3, p.362 of text For Examples see Fig. 5.5, 5.6, p.364, 365 of text

From Periodic to Aperiodic Convergence Issue given x[n] No convergence issue since the integration is over an finite interval No Gibbs phenomenon See Fig. 5.7, p.368 of text

Rectangular/Sinc 2𝜋 𝐹 𝜔 𝑛 2𝜋 2𝜋 𝐹 𝜔 𝑛 2𝜋

Fourier Transform for Periodic Signals – Unified Framework (p.16 of 4.0) Given x(t) (easy in one way)

Unified Framework: Fourier Transform for Periodic Signals (p. 17 of 4 𝑡 FS 𝑒 𝑗 𝜔 0 𝑡 𝐹 𝜔 𝜔 0 𝑥(𝑡) 2𝜋𝛿(𝜔− 𝜔 0 ) 𝑎 𝑘 𝑘𝜔 0 𝑋(𝑗𝜔) 2𝜋 𝑎 𝑘 𝛿(𝜔− 𝑘𝜔 0 ) If F

From Periodic to Aperiodic For Periodic Signals – Unified Framework Given x[n] See Fig. 5.8, p.369 of text

From Periodic to Aperiodic For Periodic Signals – Unified Framework See Fig. 5.9, p.370 of text

Signal Representation in Two Domains Time Domain Frequency Domain 2𝜋 𝑋( 𝑒 𝑗𝜔 ) 𝑥[𝑛] 𝜔 𝜔 𝛿[𝑛− 𝑘 1 ] 𝜔 1 𝜔 1 +2𝜋 𝑘 1 𝜔 𝜔 2 𝜔 2 +2𝜋 𝛿[𝑛− 𝑘 2 ] 𝑚=−∞ ∞ 2𝜋𝛿 (𝜔− 𝜔 𝑘 −2𝜋𝑚) 𝑘 2 { , −∞<𝜔<∞} {𝛿 𝑛−𝑘 , k: integer, −∞<𝑘<∞} 𝐹 𝑛 𝑒 𝑗 𝜔 𝑘 𝑛

5.2 Properties of Discrete-time Fourier Transform Periodicity Linearity

Time/Frequency Shift Conjugation

Differencing/Accumulation Time Reversal

Differentiation (p.35 of 4.0) 𝑗𝜔⋅𝑋(𝑗𝜔) = 𝜔 ⋅ 𝑋(𝑗𝜔) 𝜔 𝑋(𝑗𝜔) 𝜔 𝑋(𝑗𝜔) 𝜔 Enhancing higher frequencies De-emphasizing lower frequencies Deleting DC term ( =0 for ω=0)

Integration (p.36 of 4.0) dc term 1 𝑗 = 𝑒 −𝑗90° 1 𝑗 = 𝑒 −𝑗90° 1 𝑗𝜔 ⋅ 𝑋(𝑗𝜔) = 1 𝜔 ⋅ 𝑋(𝑗𝜔) 1 𝜔 𝑋(𝑗𝜔) 𝜔 Enhancing lower frequencies (accumulation effect) De-emphasizing higher frequencies (smoothing effect) Undefined for ω=0 Accumulation

Differencing/Accumulation 2𝜋 1 2 𝜔 1 𝜔 −2𝜋 2𝜋 Enhancing higher frequencies De-emphasizing lower freq Deleting DC term 𝑒 −𝑗𝜔 1− 𝑒 −𝑗𝜔 =2 sin ( 𝜔 2 ) Differencing/Accumulation Accumulation 2𝜋 −2𝜋 2𝜋

unique representation for orthogonal basis Time Reversal (p.29 of 3.0) the effect of sign change for x(t) and ak are identical ⋯ 𝑎 −1 𝑒 −𝑗 𝜔 0 𝑡 + 𝑎 0 + 𝑎 1 𝑒 𝑗 𝜔 0 𝑡 +⋯=𝑥(𝑡) ⋯ 𝑎 −1 𝑒 𝑗 𝜔 0 𝑡 ⋯=𝑥(−𝑡) unique representation for orthogonal basis

Time Expansion If n/k is an integer, k: positive integer See Fig. 5.13, p.377 of text See Fig. 5.14, p.378 of text

Time Expansion 𝑘𝜔 3𝜔 𝑘𝜔 3𝜔 𝑋 𝑒 𝑗𝜔 = 𝑛=−∞ ∞ 𝑥[𝑛] 𝑒 −𝑗𝜔𝑛 𝑒 −𝑗𝜔 𝑒 −𝑗3𝜔 1 𝑒 −𝑗2𝜔 𝑒 𝑗3𝜔 𝑒 −𝑗6𝜔 𝑒 𝑗𝜔 𝑛 𝑛 -1 01 2 -3 0 3 6 𝑘𝜔 3𝜔 𝑘𝜔 3𝜔 𝑋 𝑒 𝑗𝜔 = 𝑛=−∞ ∞ 𝑥[𝑛] 𝑒 −𝑗𝜔𝑛

Time Expansion Discrete-time Continuous-time = ? 𝑛 -1 0 1 2 -3 0 3 6 -3 0 3 6 Discrete-time Continuous-time 𝑥 𝑡 = 𝑛 𝑥[𝑛] 𝛿(𝑡−𝑛) 𝑥[𝑛] 𝐹 (chap4) 𝐹 (chap5) 𝑛 𝑥[𝑛] 𝑒 −𝑗𝜔𝑛 −∞ ∞ { 𝑛 𝑥 𝑛 𝛿 (𝑡−𝑛)} 𝑒 −𝜔𝑡 𝑑𝑡 𝑛 𝑥[𝑛] 𝑒 −𝑗𝜔𝑛 = 𝑡 𝑡 -1 0 1 2 -3 0 3 6 ? 𝑥 𝑎 𝑡 𝐹 1 𝑎 𝑋 ( 𝑗𝜔 𝑎 ) 𝑎= 1 𝑘 , k=integer

Differentiation in Frequency Parseval’s Relation

Multiplication Property Convolution Property frequency response or transfer function Multiplication Property periodic convolution

Input/Output Relationship (P.55 of 4.0) 𝑦(𝑡) 𝑥(𝑡) Time Domain 𝛿(𝑡) ℎ(𝑡) 𝑡 Frequency Domain 𝜔 𝜔 𝜔 1 𝜔 3 𝜔 5 𝜔 1 𝜔 3 𝜔 5

Convolution Property (p.57 of 4.0) ℎ(𝑡) 𝑌 𝑗𝜔 =𝑋(𝑗𝜔)𝐻 𝑗𝜔 𝑋 𝑗 𝜔 2 𝐻 𝑗 𝜔 2 =𝑌(𝑗 𝜔 2 ) 𝑌 𝑗𝜔 =𝑋(𝑗𝜔)𝐻 𝑗𝜔 𝑋(𝑗𝜔) 𝐻 𝑗 𝜔 2 = −∞ ∞ ℎ(𝜏) 𝑒 −𝑗 𝜔 2 𝜏 𝑑𝜏 𝐹 𝜔 𝜔 𝜔 1 𝜔 2 𝜔 1 𝜔 2 𝐻 𝑗 𝜔 1 = −∞ ∞ ℎ(𝜏) 𝑒 −𝑗 𝜔 1 𝜏 𝑑𝜏 𝑥 𝑡 ∗ℎ(𝑡)=𝑦(𝑡) 𝐹 𝐹 𝑥(𝑡) 𝑡 𝑡 ℎ(𝑡) 𝛿(𝑡) ℎ(𝑡) 𝑡 𝐹 𝑡 𝐹 𝐻(𝑗𝜔) 𝐻 𝑗 𝜔 2 𝐹 1.0 1.0 𝜔 𝜔 1 𝜔 2 1 𝜔 𝐻 𝑗𝜔 = −∞ ∞ ℎ(𝜏) 𝑒 −𝑗𝜔𝜏 𝑑𝜏 𝜔 1 𝜔 2 Transfer Function Frequency Response 𝐻 𝑗 𝜔 1 𝐻 𝑠 = −∞ ∞ ℎ(𝜏) 𝑒 −𝑠𝜏 𝑑𝜏

System Characterization Tables of Properties and Pairs See Table 5.1, 5.2, p.391, 392 of text

Vector Space Interpretation {x[n], aperiodic defined on -∞ < n < ∞}=V is a vector space basis signal sets repeats itself for very 2

Vector Space Interpretation Generalized Parseval’s Relation {X(ejω), with period 2π defined on -∞ < ω < ∞}=V : a vector space inner-product can be evaluated in either domain

Vector Space Interpretation Orthogonal Bases

Vector Space Interpretation Orthogonal Bases Similar to the case of continuous-time Fourier transform. Orthogonal bases but not normalized, while makes sense with operational definition.

Summary and Duality (p.1 of 5.0) Chap 3 Periodic Fourier Series Chap 4 Aperiodic Fourier Transform Chap 5 𝑥 𝑡 ↔𝑋 𝑗𝜔 <A> <D> 𝑥[𝑛]↔𝑋( 𝑒 𝑗𝜔 ) Continuous <C> 𝑥 𝑡 =𝑥(𝑡+𝑇) Discrete <B> 𝑥[𝑛]=𝑥[𝑛+𝑁]

5.3 Summary and Duality <A> Fourier Transform for Continuous-time Aperiodic Signals (Synthesis) (4.8) (Analysis) (4.9) -x(t) : continuous-time aperiodic in time (∆t→0) (T→∞) -X(jω) : continuous in aperiodic in frequency(ω0→0) frequency(W→∞) Duality<A> :

Case <A> (p.44 of 4.0) 𝑥 𝑡 :𝑦(𝑡) 𝑋 𝑗𝜔 :𝑧(𝜔) 𝐹 𝑇→∞ 𝑊→∞ 𝜔 𝑡 𝐹 −1 𝜔 0 →0 ∆𝑡→0 2𝜋𝑦(−𝜔) 𝑧(𝑡) 𝐹 𝜔 𝑡

<B> Fourier Series for Discrete-time Periodic Signals (Synthesis) (3.94) (Analysis) (3.95) -x[n] : discrete-time periodic in time (∆t = 1) (T = N) -ak : discrete in periodic in frequency(ω0 = 2 / N) frequency(W = 2) Duality<B> :

Case <B> Duality 𝑥[𝑛]:𝑔[𝑛] 𝑎 𝑘 :𝑓[𝑘] 𝑁 𝐹𝑆 𝑁 𝑊=2𝜋 𝑇 𝑛 𝑘 ∆𝑡=1 𝜔 0 = 2𝜋 𝑁 𝑓[𝑛] 𝑁 1 𝑁 𝑔[−𝑘] 𝑁 𝐹𝑆 𝑛 𝑘

<C> Fourier Series for Continuous-time Periodic Signals (Synthesis) (3.38) (Analysis) (3.39) -x(t) : continuous-time periodic in time (∆t → 0) (T = T) -ak : discrete in aperiodic in frequency(ω0 = 2 / T) frequency(W → ∞)

Case <C> <D> Duality 𝑎 𝑘 :𝑔[𝑘] 𝑇 𝑊→∞ 𝑥(𝑡):𝑣(𝑡) 𝐹𝑆 𝑦[−𝑘] 𝑇 𝑧(𝑡) 𝑡 𝑘 0 1 2 3 𝜔 0 = 2𝜋 𝑇 ∆𝑡→0 𝑋 𝑒 𝑗𝜔 :𝑧(𝜔) <D> 𝑇→∞ 2𝜋 𝑣[−𝜔] 𝐹 𝑥[𝑛]:𝑦[𝑛] 𝑛 𝜔 𝑔[𝑛] ∆𝑡=1 𝜔 0 →0 For <C> For <D> Duality

<D> Discrete-time Fourier Transform for Discrete-time Aperiodic Signals (Synthesis) (5.8) (Analysis) (5.9) -x[n] : discrete-time aperiodic in time (∆t = 1) (T→∞) -X(ejω) : continuous in periodic in frequency(ω0→0) frequency(W = 2)

Duality<C> / <D> For <C> For <D> Duality taking z(t) as a periodic signal in time with period 2, substituting into (3.38), ω0 = 1 which is of exactly the same form of (5.9) except for a sign change, (3.39) indicates how the coefficients ak are obtained, which is of exactly the same form of (5.8) except for a sign change, etc. See Table 5.3, p.396 of text

𝒏

More Duality Discrete in one domain with ∆ between two values → periodic in the other domain with period   Continuous in one domain (∆ → 0) → aperiodic in the other domain  

Harmonically Related Exponentials for Periodic Signals (P.11 of 3.0) 𝑉={𝑥(𝑡) 𝑥(𝑡) periodic, fundamental period [𝑛] =𝑇(𝑁)} T (𝑁) 𝑡, 𝑛 𝑎 4 𝑡, 𝑛 𝑎 1 𝑎 3 𝑎 5 𝑎 0 𝑘=1 𝑘 𝜔 0 = 2𝜋 𝑇 (𝑁) 0 1 2 3 4 5 𝑡, 𝑛 𝜔 𝑘=2 𝜔 0 3𝜔 0 5𝜔 0 𝑡, 𝑛 2𝜔 0 4𝜔 0 𝑘=3 All with period T: integer multiples of ω0 Discrete in frequency domain (𝑁)

Extra Properties Derived from Duality examples for Duality <B> duality duality

Unified Framework Fourier Transform : case <A> (4.8) (4.9)

Unified Framework Discrete frequency components for signals periodic in time domain: case <C> you get (3.38) (applied on (4.8)) Case <C> is a special case of Case <A>

Unified Framework: Fourier Transform for Periodic Signals (p. 17 of 4 𝑡 FS 𝑒 𝑗 𝜔 0 𝑡 𝐹 𝜔 𝜔 0 𝑥(𝑡) 2𝜋𝛿(𝜔− 𝜔 0 ) 𝑎 𝑘 𝑘𝜔 0 𝑋(𝑗𝜔) 2𝜋 𝑎 𝑘 𝛿(𝜔− 𝑘𝜔 0 ) If F

Unified Framework Discrete time values with spectra periodic in frequency domain: case <D> (4.9) becomes (5.9) Case <D> is a special case of Case <A> Note : ω in rad/sec for continuous-time but in rad for discrete-time

Time Expansion (p.41 of 5.0) Discrete-time Continuous-time = ? 𝑛 -1 0 1 2 -3 0 3 6 Discrete-time Continuous-time 𝑥 𝑡 = 𝑛 𝑥[𝑛] 𝛿(𝑡−𝑛) 𝑥[𝑛] 𝐹 (chap4) 𝐹 (chap5) 𝑛 𝑥[𝑛] 𝑒 −𝑗𝜔𝑛 −∞ ∞ { 𝑛 𝑥 𝑛 𝛿 (𝑡−𝑛)} 𝑒 −𝜔𝑡 𝑑𝑡 𝑛 𝑥[𝑛] 𝑒 −𝑗𝜔𝑛 = 𝑡 𝑡 -1 0 1 2 -3 0 3 6 ? 𝑥 𝑎 𝑡 𝐹 1 𝑎 𝑋 ( 𝑗𝜔 𝑎 ) 𝑎= 1 𝑘 , k=integer

Unified Framework Both discrete/periodic in time/frequency domain: case <B> -- case <C> plus case <D> periodic and discrete, summation over a period of N (4.8) becomes (4.9) becomes (3.94) (3.95)

Unified Framework Cases <B> <C> <D> are special cases of case <A> Dualities <B>, <C>/<D> are special case of Duality <A> Vector Space Interpretation ----similarly unified

Summary and Duality (p.1 of 5.0) Chap 3 Periodic Fourier Series Chap 4 Aperiodic Fourier Transform Chap 5 𝑥 𝑡 ↔𝑋 𝑗𝜔 <A> <D> 𝑥[𝑛]↔𝑋( 𝑒 𝑗𝜔 ) Continuous <C> 𝑥 𝑡 =𝑥(𝑡+𝑇) Discrete <B> 𝑥[𝑛]=𝑥[𝑛+𝑁]

Examples Example 5.6, p.371 of text

Examples Example 4.8, p.299 of text (P.76 of 4.0) Discrete (∆𝑡=𝑇) → Periodic Periodic ( 𝑇=𝑇) → Discrete (𝑊= 2𝜋 𝑇 ) ( 𝜔 0 = 2𝜋 𝑇 )

Examples Example 5.11, p.383 of text   time shift property

Examples Example 5.14, p.387 of text

Examples Example 5.17, p.395 of text

Examples Example 3.5, p.193 of text (P. 58 of 3.0) (a) (b) (c)

Rectangular/Sinc (p.21 of 5.0)

Problem 5.36(c), p.411 of text

Problem 5.43, p.413 of text

Problem 5.46, p.415 of text

Problem 5.56, p.422 of text

Problem 3.70, p.281 of text 2-dimensional signals (P. 65 of 3.0) 𝑒 𝑗 𝜔 1 𝑡 1 𝑒 𝑗 𝜔 2 𝑡 2 𝑡 2 𝑡 2 𝑡 1 𝑡 1 𝑡 2 𝑒 𝑗( 𝜔 1 𝑡 1 + 𝜔 2 𝑡 2 ) 𝑡 1

Problem 3.70, p.281 of text 2-dimensional signals (P. 64 of 3.0) 𝜔 10 𝑡 2 𝐹𝑆 ( 𝑡 2 ) 𝑇 1 𝜔 1 𝑇 2 𝑡 1 𝑡 2 𝑡 1 𝜔 1 different 𝑡 2 𝜔 2 𝑡 2 𝐹𝑆 ( 𝜔 1 ) (0, 𝜔 2 ) ( 𝜔 1 , 𝜔 2 ) 𝜔 1 𝜔 20 ( 𝜔 1 ,0) 𝜔 10

An Example across Cases <A><B><C><D> <C> 𝑥 𝛼𝑡 𝐹𝑆 𝑎 𝑘 (Sec. 3.5.4) ( 𝑇 ′ =𝑇/𝛼, 𝜔 0 ′ =𝛼 𝜔 0 ) (Table 3.1) <B> 𝑥 (𝑚) [𝑛] 𝐹𝑆 1 𝑚 𝑎 𝑘 (Table 3.2)

Time/Frequency Scaling (p.38 of 4.0) (time reversal) 𝑡 𝐹 𝜔 𝑥(𝑡) 𝑥 𝑎𝑡 , 𝑎>1 𝑥 𝑎𝑡 , 𝑎<1 𝑋(𝑗𝜔) 1 𝑎 𝑋 𝑗 𝜔 𝑎 , 𝑎>1 1 𝑎 𝑋 𝑗 𝜔 𝑎 , 𝑎<1 See Fig. 4.11, p.296 of text

Single Frequency (p.40 of 4.0) cos 𝜔 0 𝑡 = 1 2 𝑒 𝑗 𝜔 0 𝑡 + 𝑒 −𝑗 𝜔 0 𝑡 𝐹 𝜔 𝑡 −𝜔 0 𝜔 0 𝜔 𝐹 𝑡 −2𝜔 0 2𝜔 0 𝜔 𝐹 𝑡 − 1 2 𝜔 0 1 2 𝜔 0

Parseval’s Relation (p.37 of 4.0) total energy: energy per unit time integrated over the time total energy: energy per unit frequency integrated over the frequency 𝐴 = 𝑖 𝑎 𝑖 𝑣 𝑖 = 𝑘 𝑏 𝑘 𝑢 𝑘 𝐴 2 = 𝑖 𝑎 𝑖 2 = 𝑘 𝑏 𝑘 2

Single Frequency <A> 𝑥 𝑎𝑡 𝐹 1 𝑎 𝑋( 𝑗𝜔 𝑎 ) (4.34) 𝑥 𝑡 = cos 𝜔 0 𝑡 𝐹 𝑋 𝑗𝜔 =𝜋𝛿 𝜔− 𝜔 0 +⋯ 𝑥 𝑎𝑡 = cos 𝑎 𝜔 0 𝑡 𝐹 1 𝑎 𝜋 𝛿 𝜔 𝑎 − 𝜔 0 +⋯ 𝑎⋅ −∞ ∞ 𝛿 𝜔 𝑎 𝑑 𝜔 𝑎 =𝑎 𝛿 𝜔 𝑎 =𝑎𝛿 𝜔 𝛿 𝜔 𝑎 − 𝜔 0 =𝑎 𝛿 𝜔−𝑎 𝜔 0 =1 𝑥 𝑎𝑡 = cos 𝑎 𝜔 0 𝑡 𝐹 1 𝑎 𝑋 ( 𝑗𝜔 𝑎 )=𝜋𝛿 𝜔−𝑎 𝜔 0 +⋯

Another Example See Figure 4.14 (example 4.8), p.300 of text 𝑋 𝑗𝜔 𝑥 𝑡 2𝜋 𝑇 𝐹 ⋯ ⋯ ⋯ ⋯ 𝜔 𝑡 2𝜋 𝑇 𝑇 See Figure 4.14 (example 4.8), p.300 of text ⋯ 𝐹 1 2𝜋 2𝑇 ⋯ ⋯ ⋯ ⋯ 𝑡 𝜔 2𝑇 2𝜋 2𝑇

Cases <C><D> 𝛼: any real number 𝑇 𝐹𝑆 𝑡 𝜔 𝜔 0 = 2𝜋 𝑇 𝛼>1 𝑡 𝜔 𝜔 0 = 2𝜋 𝑇/𝛼 𝑇/𝛼 𝑥 𝛼𝑡 𝐹𝑆 𝑎 𝑘 ( 𝑇 ′ =T/α , 𝜔 0 ′ =α 𝜔 0 ) Duality 2𝜋 <D> 𝑘: positive integer only 𝜔 𝑛 2𝜋 𝐹 𝑛 𝜔 𝑥 (𝑘) [𝑛] 𝐹 𝑋( 𝑒 𝑗𝑘𝜔 )

Cases <B> 𝑥 (𝑚) 𝑛 𝐹𝑆 1 𝑚 𝑎 𝑘 𝑥 (𝑚) 𝑛 𝐹𝑆 1 𝑚 𝑎 𝑘 2𝜋 𝑁 𝑁 𝐹𝑆 𝑎 𝑘 𝑥 𝑛 𝑛 𝜔 2𝜋 𝑁 𝑥 (2) 𝑛 2𝜋 𝑚𝑁 𝜔 𝑚𝑁 2𝜋 𝑚𝑁 2𝜋 𝑚 𝑚 ∥ 2 ∥ 2 𝑥 (𝑚) 𝑛 𝐹𝑆 𝑎 𝑘 ′ = 1 𝑚𝑁 𝑛=<𝑚𝑁> 𝑥 (𝑚) 𝑛 𝑒 −𝑗𝑘 2𝜋 𝑚𝑁 𝑛 𝑙𝑚 𝑙𝑚 𝑙=<𝑁> = 1 𝑚 ⋅[ 1 𝑁 𝑙=<𝑁> 𝑥 𝑙 𝑒 −𝑗𝑘 2𝜋 𝑁 𝑙 ] = 1 𝑚 ⋅ 𝑎 𝑘