What does the discrete-time Fourier transform (DTFT) represent for a finite sequence?

A continuous, periodic function of frequency; the DFT samples one cycle at equally spaced frequencies.

How does sample spacing determine DFT frequency spacing?

For sampling interval $\Delta t$ and $N$ samples, the spacing is $\Delta f=1/(N\Delta t)$.

What does Parseval's theorem say for the unscaled-forward DFT convention?

The sample-domain inner product equals the frequency-domain inner product multiplied by $1/N$.

What is the effect of unitary DFT normalization?

Using $1/\sqrt{N}$ in both directions makes the transform preserve inner products and vector lengths.

What is spectral leakage?

The spreading of a signal component's energy across multiple frequency bins when analyzing a finite record.

Signal processingLesson 2 of 4 in the course Fourier analysis

Discrete Fourier Transform: Properties, Interpretation, and Applications

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Summary

The discrete Fourier transform (DFT) changes a finite list of samples into a description of the oscillations that make up those samples. This frequency-domain view is useful for analyzing signals, images, and periodic data.

$X_k=\sum_{n=0}^{N-1}x_n e^{-i2\pi kn/N},\qquad k=0,\ldots,N-1.$

Each coefficient measures how strongly a particular discrete complex sinusoid is represented in the input. Its magnitude describes the component's strength, while its complex phase describes its phase offset. The index $k$ identifies a frequency of $k$ cycles per $N$ samples.

All flashcards

What does the discrete-time Fourier transform (DTFT) represent for a finite sequence?: A continuous, periodic function of frequency; the DFT samples one cycle at equally spaced frequencies.
How does sample spacing determine DFT frequency spacing?: For sampling interval $\Delta t$ and $N$ samples, the spacing is $\Delta f=1/(N\Delta t)$ .
What does Parseval's theorem say for the unscaled-forward DFT convention?: The sample-domain inner product equals the frequency-domain inner product multiplied by $1/N$ .
What is the effect of unitary DFT normalization?: Using $1/\sqrt{N}$ in both directions makes the transform preserve inner products and vector lengths.
What is spectral leakage?: The spreading of a signal component's energy across multiple frequency bins when analyzing a finite record.

Quiz questions

For $N$ samples taken at intervals of $\Delta t$ , what is the spacing between adjacent DFT frequency bins?Answer: $1/(N\Delta t)$
What does the circular convolution theorem say about the DFT of two length- $N$ sequences' circular convolution?Answer: It equals the pointwise product of their DFTs.
If a length- $N$ input sequence is real-valued, what symmetry do its DFT coefficients satisfy?Answer: $X_k=X_{(-k)\bmod N}^*$
Why can zero-padding be used when computing linear convolution with a length- $N$ DFT?Answer: Sufficient padding prevents circular wraparound from corrupting the full linear-convolution result.
How can a two-dimensional DFT of an image be computed using one-dimensional transforms?Answer: Transform each row and then each column; reversing the order gives the same result.

Based on the Wikipedia article Discrete Fourier transform, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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