Fourier Transforms for Continuous/Discrete Time/Frequency

The Fourier transform can be defined for signals which are

• discrete or continuous in time, and
• finite or infinite in duration.

This results in four cases. As you might expect, the frequency domain has the same cases:

• discrete or continuous in frequency, and
• finite or infinite in bandwidth.

When time is discrete, the frequency axis is finite, and vice versa.

Reference [243] develops the DFT in detail--the discrete-time, discrete-frequency case. In the DFT, both the time and frequency axes are finite in length. Table 2.1 below summarizes the four Fourier-transform cases corresponding to discrete or continuous time and/or frequency. They are discussed further in [243].

In all four cases, the Fourier transform can be interpreted as the inner product of the signal with a complex sinusoid at radian frequency [243]:

where is appropriately adapted, e.g.,

• (Fourier transform case),
• (DTFT case),
• (DFT case).

In spectral modeling of audio, we usually deal with indefinitely long signals. Fourier analysis of an indefinitely long discrete-time signal is carried out using the Discrete Time Fourier Transform (DTFT). In practical situations we can only deal with finite-duration signals, so really we will always use the Discrete Fourier Transform (DFT) [243].

In the remainder of this chapter, the DTFT is defined and selected Fourier theorems are stated and proved for the DTFT case. Additionally, the Fourier Transform (FT) is defined, and selected FT theorems are stated and proved as well. The theorems for the DFT case are detailed in [243].

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Subsections

• Discrete Time Fourier Transform
• Fourier Transform (FT) and Inverse
  • Existence of the Fourier Transform
  
  
• Fourier Theorems for the DTFT
  • Linearity of the DTFT
  • Time Reversal
  • Symmetry of the DTFT for Real Signals
    • Real Even (or Odd) Signals
  • Shift Theorem
  • Convolution Theorem
  • Correlation Theorem
  • Autocorrelation
  • Power Theorem
  • Stretch Operator
  • Repeat (Scaling) Operator
  • Stretch/Repeat (Scaling) Theorem
  • Downsampling and Aliasing
    • Proof of Aliasing Theorem
  • Differentiation Theorem Dual
  
  
• Continuous Fourier Theorems
  • Radians versus Cycles
  • Differentiation Theorem
  • Differentiation Theorem Dual
  • Scaling Theorem
  • Shift Theorem
  • Convolution Theorem
  • Flip Theorems
  • Power Theorem
  • The Continuous-Time Impulse
  • Gaussian Pulse
  • Rectangular Pulse
  • Sinc Impulse
  • Impulse Trains
  • Poisson Summation Formula
  • Sampling Theory
  • The Uncertainty Principle
    • Second Moments
    • Time-Limited Signals
    • Time-Bandwidth Products are Unbounded Above
  
  
• Relation of Smoothness to Roll-Off Rate
• Exercises

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Order a Hardcopy of Spectral Audio Signal Processing

Previous: Tighter Bounds for Minimum Window Length
Next: Discrete Time Fourier Transform (DTFT)

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.