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.

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

Reference [264] 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 (next page) summarizes the four Fourier-transform cases corresponding to discrete or continuous time and/or frequency.

In all four cases, the Fourier transform can be interpreted as
the *inner product* of the signal
with a complex sinusoid at
radian frequency
[264]:

(3.1) |

where is appropriately adapted,

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).^{3.1}Below, the DTFT is defined, and selected Fourier theorems are stated
and proved for the DTFT case. Additionally, for completeness, the
Fourier Transform (FT) is defined, and selected FT theorems are stated
and proved as well. Theorems for the DFT case are detailed in
[264].^{3.2}

- Discrete Time Fourier Transform
- Fourier Transform (FT) and
Inverse

- Fourier Theorems for the DTFT
- Linearity of the DTFT
- Time Reversal
- Symmetry of the DTFT for Real Signals
- Shift Theorem
- Convolution Theorem
- Correlation Theorem
- Autocorrelation
- Power Theorem
- Stretch Operator
- Repeat (Scaling) Operator
- Stretch/Repeat (Scaling) Theorem
- Downsampling and Aliasing
- Differentiation Theorem Dual

- Continuous-Time Fourier Theorems

- Spectral Interpolation

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University