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Now we can do something similar with the two 2-point DFT's to get equations for a; for j = 0,1,2,3: (DFT (αo, α2)o + DFT (αo, α2)1) (DFT (ao, a₂)o-DFT (αo, α₂)1) (DFT (α₁, α3)0 + DFT (α₁, α3)1) -(DFT (a1, a2)3 — DFT (α₁, α3)1). Exercise 1. Using the definition of DFT in Equation (3), prove Equations (11). αo 4₂= a₁ =

Now we can do something similar with the two 2-point DFT's to get equations for a; for j = 0,1,2,3: (DFT (αo, α2)o + DFT (αo, α2)1) (DFT (ao, a₂)o-DFT (αo, α₂)1) (DFT (α₁, α3)0 + DFT (α₁, α3)1) -(DFT (a1, a2)3 — DFT (α₁, α3)1). Exercise 1. Using the definition of DFT in Equation (3), prove Equations (11). αo 4₂= a₁ =

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 30E
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Please do Exercise 1 and please show step by step and explain

N-1 S; = Σ axelnikj/N k=0 1 N-1 an Σsje-2nijn/N j=0 where these equations hold for any sequence of N complex numbers ao, ₁,...N-1. In fact, instead of starting with the a's we could start with an arbitrary sequence of complex numbers So,... SN-1 and define the sequence {a} in terms of {s;} by using Equation (2). We can then recover the sj's from these an's from Equation (1). (Note we have written s; instead of s(j) to emphasize the remarkably symmetric situation between the two sequences {an} in terms of {s;}.)' N-1 IDFT (So,... SN-1)n == In fact, what we have just derived is the complex form of the famous discrete Fourier transform, abbrevi- ated as DFT. Equation (1) expresses that the sequence {s;} is the DFT of {a}, while Equation (1) expresses that the sequence {n} is the inverse discrete Fourier transform (abbreviated IDFT) of {s;}. For later con- venience, we will introduce the notation: DFT (ao,...,N-1);= Σ²kj/N, k=0 1 N-1 ΣSje-2mijn/N (1) j=0,..., N-1 (2) n = 0,..., N-1 (3) (4) The subscripts j and n on the left-hand sides of these equations reflects the fact that the DFT (or IDFT) of a sequence of N complex numbers is itself a sequence of N complex numbers. The sums on the right-hand sides are the formulas for the different components of the DFT and IDFT. With this notation, we may write Equations (1) and (2) very succinctly as: {sj} = DFT({aj}) → {aj} = IDFT({sj}), (5) where {s;} and {a;} denote the sequences So,... SN-1 and ao... N-1, respectively. The notation DFT and IDFT recalls the notion of function and inverse function: in fact, the N-point DFT may be considered as a function from CN to CN, and the N-point IDFT is its inverse. It is impossible to underestimate the importance of the DFT in modern science, engineering, and math- ematics. It pops up everywhere, even in number theory!
Transcribed Image Text:N-1 S; = Σ axelnikj/N k=0 1 N-1 an Σsje-2nijn/N j=0 where these equations hold for any sequence of N complex numbers ao, ₁,...N-1. In fact, instead of starting with the a's we could start with an arbitrary sequence of complex numbers So,... SN-1 and define the sequence {a} in terms of {s;} by using Equation (2). We can then recover the sj's from these an's from Equation (1). (Note we have written s; instead of s(j) to emphasize the remarkably symmetric situation between the two sequences {an} in terms of {s;}.)' N-1 IDFT (So,... SN-1)n == In fact, what we have just derived is the complex form of the famous discrete Fourier transform, abbrevi- ated as DFT. Equation (1) expresses that the sequence {s;} is the DFT of {a}, while Equation (1) expresses that the sequence {n} is the inverse discrete Fourier transform (abbreviated IDFT) of {s;}. For later con- venience, we will introduce the notation: DFT (ao,...,N-1);= Σ²kj/N, k=0 1 N-1 ΣSje-2mijn/N (1) j=0,..., N-1 (2) n = 0,..., N-1 (3) (4) The subscripts j and n on the left-hand sides of these equations reflects the fact that the DFT (or IDFT) of a sequence of N complex numbers is itself a sequence of N complex numbers. The sums on the right-hand sides are the formulas for the different components of the DFT and IDFT. With this notation, we may write Equations (1) and (2) very succinctly as: {sj} = DFT({aj}) → {aj} = IDFT({sj}), (5) where {s;} and {a;} denote the sequences So,... SN-1 and ao... N-1, respectively. The notation DFT and IDFT recalls the notion of function and inverse function: in fact, the N-point DFT may be considered as a function from CN to CN, and the N-point IDFT is its inverse. It is impossible to underestimate the importance of the DFT in modern science, engineering, and math- ematics. It pops up everywhere, even in number theory!
Now we can do something similar with the two 2-point DFT's to get equations for a; for j = 0,1,2,3: (DFT (αo, a2)o + DFT (αo, α2)1) DFT (α0, α₂)1) (DFT (a₁, a3)0 + DFT (α₁, α3)1) (DFT (a1, a2)3 – DFT (α₁, α3)1). Exercise 1. Using the definition of DFT in Equation (3), prove Equations (11). α = = 02 (DFT (0, 2)0 2 α₁ = α3 = (11)
Transcribed Image Text:Now we can do something similar with the two 2-point DFT's to get equations for a; for j = 0,1,2,3: (DFT (αo, a2)o + DFT (αo, α2)1) DFT (α0, α₂)1) (DFT (a₁, a3)0 + DFT (α₁, α3)1) (DFT (a1, a2)3 – DFT (α₁, α3)1). Exercise 1. Using the definition of DFT in Equation (3), prove Equations (11). α = = 02 (DFT (0, 2)0 2 α₁ = α3 = (11)
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