Discrete Fourier Transformation (DFT) Prof. Seungchul Lee - - PowerPoint PPT Presentation

Discrete Fourier Transformation (DFT)

• Prof. Seungchul Lee

Industrial AI Lab.

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Eigen-Analysis

(System or Linear Transformation)

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Eigenvector and Eigenvalues

• Given matrix 𝐵
• Eigenvectors 𝑤 are input signals that emerge at the system output unchanged (except for a scaling by

the eigenvalue 𝜇𝑙) and so are somehow “fundamental” to the system

• Using this, we can find the following equation
• We can change to

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Eigen-analysis of LTI Systems (Finite-Length Signals)

• For length-𝑂 signals, 𝐼 is an 𝑂 × 𝑂 circulent matrix with entries

where ℎ is the impulse response

• Goal: calculate the eigenvectors and eigenvalues of 𝐼

– Fact: the eigenvectors of a circulent matrix (LTI system) are the complex harmonic sinusoids – The eigenvalue 𝜇𝑙 ∈ ℂ corresponding to the sinusoid eigenvectors 𝑡𝑙 is called the frequency response at frequency 𝑙 since it measures how the system “responds” to 𝑡𝑙

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Eigenvector of LTI Systems (Finite-Length Signals)

• Prove that

– harmonic sinusoids are the eigenvectors of LTI systems simply by computing the circular convolution with input 𝑡𝑙 and applying the periodicity of the harmonic sinusoids

• 𝜇𝑙 means the number of 𝑡𝑙 in ℎ[𝑜] ⇒ similarity

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Eigenvector Matrix of Harmonic Sinusoids

• Stack 𝑂 normalized harmonic sinusoid 𝑡𝑙 𝑙=0

𝑂−1 as columns into an 𝑂 × 𝑂 complex orthonormal basis

matrix

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Signal Decomposition by Harmonic Sinusoids

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Basis

• A basis {𝑐𝑙} for a vector space 𝑊 is a collection of vectors from 𝑊 that linearly independent and span 𝑊
• Basis matrix: stack the basis vectors 𝑐𝑙 as columns
• Using this matrix 𝐶, we can now write a linear combination of basis elements as the matrix/vector

product

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Orthonormal Basis

• An orthogonal basis 𝑐𝑙 𝑙=0

𝑂−1 for a vector space 𝑊

– a basis whose elements are mutually orthogonal

• An orthonormal basis 𝑐𝑙 𝑙=0

𝑂−1 for a vector space 𝑊

– a basis whose elements are mutually orthogonal and normalized in the 2-norm

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Orthonormal Basis

• 𝐶 is a unitary matrix

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Signal Represented by Orthonormal Basis

• Signal representation by orthonormal basis 𝑐𝑙 𝑙=0

𝑂−1 and orthonormal basis matrix 𝐶

• Synthesis: build up the signal 𝑦 as a linear combination of the basis elements 𝑐𝑙 weighted by the

weights 𝛽𝑙

• Analysis: compute the weights 𝛽𝑙 such that the synthesis produces 𝑦; the weights 𝛽𝑙 measures the

similarity between 𝑦 and the basis element 𝑐𝑙

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Harmonic Sinusoids are an Orthonormal Basis

• Stack 𝑂 normalized harmonic sinusoid 𝑡𝑙 𝑙=0

𝑂−1 as columns into an 𝑂 × 𝑂 complex orthonormal basis

matrix

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Discrete Fourier Transform (DFT)

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DFT and Inverse DFT

• Jean Baptiste Joseph Fourier had the radical idea of proposing that all signals

could be represented as a linear combination of sinusoids

• Analysis (Forward DFT)

– The weight 𝑌[𝑙] measures the similarity between 𝑦 and the harmonic sinusoid 𝑡𝑙 – It finds the “frequency contents” of 𝑦 at frequency 𝑙

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DFT and Inverse DFT

• Jean Baptiste Joseph Fourier had the radical idea of proposing that all signals

could be represented as a linear combination of sinusoids

• Synthesis (Inverse DFT)

– It is returning to time domain – It builds up the signal 𝑦 as a linear combination of 𝑡𝑙 weighted by the 𝑌[𝑙]

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Unnormalized DFT

• Normalized forward and inverse DFT
• Unnormalized forward and inverse DFT

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Harmonic Sinusoids are an Orthonormal Basis

• Stack 𝑂 normalized harmonic sinusoid 𝑡𝑙 𝑙=0

𝑂−1 as columns into an 𝑂 × 𝑂 complex orthonormal basis

matrix

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Eigen-decomposition and Diagonalization

• 𝐼 is circulent LTI System matrix
• 𝑇 is harmonic sinusoid eigenvectors matrix (corresponds to DFT/IDFT)
• Λ is eigenvalue diagonal matrix (frequency response)
• The eigenvalues are the frequency response (unnormalized DFT of the impulse response)
• Place the 𝑂 eigenvalues 𝜇𝑙 𝑙=0

𝑂−1 on the diagonal of an 𝑂 × 𝑂 matrix

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Eigen-decomposition and Diagonalization

• 𝐼 is circulent LTI System matrix
• 𝑇 is harmonic sinusoid eigenvectors matrix (corresponds to DFT/IDFT)
• Λ is eigenvalue diagonal matrix (frequency response)

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Eigen-decomposition and Diagonalization

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Eigen-decomposition and Diagonalization

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Eigen-decomposition and Diagonalization

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Eigen-decomposition and Diagonalization

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DFT in MATLAB

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DFT in MATLAB

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DFT Function

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Example: DFT

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Example: DFT

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Example: DFT

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Example: DFT

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Fast Fourier Transform (FFT)

• FFT algorithms are so commonly employed to compute DFT that the term 'FFT' is often used to mean

'DFT'

– The FFT has been called the "most important computational algorithm of our generation" – It uses the dynamic programming algorithm (or divide and conquer) to efficiently compute DFT.

• DFT refers to a mathematical transformation or function, whereas 'FFT' refers to a specific family of

algorithms for computing DFTs.

– use fft command to compute dft – fft (computationally efficient)

• We will use the embedded fft function without going too much into detail.

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DFT Properties

• DFT pair
• DFT Frequencies

– 𝑌[𝑙] measures the similarity between the time signal 𝑦[𝑜] and the harmonic sinusoid 𝑡𝑙[𝑜] – 𝑌[𝑙] measures the “frequency content” of 𝑦[𝑜] at frequency 𝜕𝑙 =

2𝜌 𝑂 𝑙

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DFT Properties

• DFT and Circular Shift

– No amplitude changed – Phase changed

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DFT Properties

• DFT and Modulation

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DFT Properties

• DFT and Circular Convolution

– Circular convolution in the time domain = multiplication in the frequency domain

• Proof

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Filtering in Frequency Domain

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• Circular convolution in the time domain = multiplication in the frequency domain

Example: Low-Pass Filter

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Example: High-Pass Filter

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Filtering in Time Domain

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Filtering in Frequency Domain

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