Lecture 13: Practical Fourier Transforms Foundations of Digital Signal Processing Outline • The Discrete Fourier Transform (DFT) • Circular Convolution • The DTFT and the DFT: The Relationship • The Fast Fourier Transform Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 1 News Homework #5 . Due today . Submit via canvas Coding Problem #4 . Due today . Submit via canvas Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 2 Lecture 13: Practical Fourier Transforms Foundations of Digital Signal Processing Outline • The Discrete Fourier Transform (DFT) • Circular Convolution • The DTFT and the DFT: The Relationship • The Fast Fourier Transform Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 3 Deriving Transforms Consider the Inverse Discrete-Time Fourier Transform…. What happens if we sample X ? 1 = +n � 2 2 Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 4 Deriving Transforms Consider the Inverse Discrete-Time Fourier Transform…. What happens if we sample X ? 1 = +n � 1 2 2 = ∞ +n � 2 � − 2 2 =−∞ Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 5 Deriving Transforms Consider the Inverse Discrete-Time Fourier Transform…. What happens if we sample X ? 1 = +n � 1 2 2 = 2 ∞ +n � 2 � − 2 0 =−∞ 0 < 0 ≥ < 2 ≥ 2 Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 6 Deriving Transforms Consider the Inverse Discrete-Time Fourier Transform…. What happens if we sample X ? 1 = +n � 1 2 2 = 2 ∞ +n � 2 � − 2 0 =−∞ 2 Let 2 / = −1 = = 2 / +n � =0 Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 7 Deriving Transforms Consider the Inverse Discrete-Time Fourier Transform…. What happens if we sample X ? 1 = +n � 1 2 2 = 2 ∞ +n � 2 � − 2 0 =−∞ = −1 Let 2 / = 2 + n = 2 / � =0 Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 8 Deriving Transforms Consider the Inverse Discrete-Time Fourier Transform…. What happens if we sample X ? = −1 2 + kn � =0 . The DTFT becomes the Discrete-Time Fourier Series Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 9 The Discrete Fourier Transform The Discrete-Time Fourier Series . Analysis Equations 1 = −1 2 − � . Synthesis Equations =0 = −1 2 + � =0 Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 10 The Discrete Fourier Transform The Discrete Fourier Transform (DFT) . Analysis Equations = −1 2 − � . Synthesis Equations =0 1 = −1 2 + � =0 Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 11 The Discrete Fourier Transform The Discrete Fourier Transform (DFT) . Analysis Equations = −1 2 − � . Synthesis Equations =0 1 = −1 2 + � Question: What are the properties=0 of the DFT? . How does this work? Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 12 The Discrete Fourier Transform The Discrete Fourier Transform (DFT) . Analysis Equations = −1 2 − � . Synthesis Equations =0 1 = −1 2 + � Question: What are the properties=0 of the DFT? . How does this work? Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 13 The Discrete Fourier Transform Properties of the Discrete Fourier Transform . If is real ◊ real is even ◊ imag is odd ◊ is even ◊ is odd ∠ Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 14 The Discrete Fourier Transform Properties of the Discrete Fourier Transform . If is real and odd real = 0 ◊ ◊ imag is odd ◊ is even ◊ =0 ∠ Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 15 The Discrete Fourier Transform Properties of the Discrete Fourier Transform . If is real and even ◊ real is even ◊ imag = 0 ◊ is even ◊ =0 ∠ Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 16 The Discrete Fourier Transform Circular Convolution . Multiplication property (DFT) 1 . ↔ ⊛ Convolution property (DFT) ⊛ ↔ Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 17 The Discrete Fourier Transform Circular Convolution . Multiplication property (DTFT) 1 . ↔ ⊛ Convolution property (DTFT2) ∗ ↔ Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 18 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ⊛ Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 19 Circular Convolution What is Circular Convolution? . Convolution for periodic signals . Convolve ◊ One period of one signal ◊ With the entire second signal Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 20 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ◊ One period of one signal ◊ With the entire second signal 4 4 − ⊛ 2 2 − Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 21 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ◊ One period of one signal ◊ With the entire second signal 4 4 − ∗ 2 2 − Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 22 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ◊ One period of one signal ◊ With the entire second signal 4 4 − 2 2 − Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 23 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ◊ One period of one signal ◊ With the entire second signal 2 2 − ⊛ 2 2 − Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 24 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ◊ One period of one signal ◊ With the entire second signal 2 2 − ∗ 2 2 − Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 25 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ◊ One period of one signal ◊ With the entire second signal 2 2 − ⊛ Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 26 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ◊ OR Perform convolution between two periods ◊ But assume periodic boundary conditions 4 4 − ⊛ 2 2 − Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 27 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ◊ OR Perform convolution between two periods ◊ But assume periodic boundary conditions 4 4 − 2 2 − Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 28 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ◊ OR Perform convolution between two periods ◊ But assume periodic boundary conditions 2 2 − ⊛ Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 29 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ◊ OR Perform convolution between two periods ◊ But assume periodic boundary conditions 2 2 − Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 30 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ◊ One period of one signal ◊ With the entire second signal = ⊛ 6 Assume these are finiteℎlength -8 signals − Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 31 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ◊ One period of one signal ◊ With the entire second signal = ∗ 6 Assume these are finiteℎlength -8 signals − Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 32 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ◊ One period of one signal ◊ With the entire second signal Assume these are finite length-8 signals Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 33 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ◊ One period of one signal ◊ With the entire second signal Only consider 8 samples Assume these are finite length-8 signals Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 34 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ◊ One period of one signal ◊ With the entire second signal Assume these are finite length-8 signals Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 35 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ◊ OR Perform convolution between two periods ◊ But assume periodic boundary conditions = ⊛ 6 Assume these are finiteℎlength -8 signals − Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 36 Circular Convolution What is Circular Convolution? . Convolution for periodic signals ◊ OR Perform convolution between two periods ◊ But assume periodic boundary conditions Assume these are finite length-8 signals Foundations of Digital Signal Processing Lecture 13: Practical Fourier Transforms 37 Circular