ECE 8443 – Pattern RecognitionEE 3512 – Signals: Continuous and Discrete

• Objectives:Derivation of the DFTRelationship to DTFTDFT of Truncated SignalsTime Domain Windowing

• Resources:Wiki: Discrete Fourier TransformWolfram: Discrete Fourier TransformDSPG: The Discrete Fourier TransformWiki: Time Domain WindowsISIP: Java Applet

LECTURE 17: DISCRETE FOURIER TRANSFORM

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EE 3512: Lecture 17, Slide 2

• The Discrete-Time Fourier Transform:

• Not practical for (real-time) computation on a digital computer.

• Solution: limit the extent of the summation to N points and evaluate the continuous function of frequency at N equispaced points:

• MATLAB code for the DFT:

• The exponentials can be precomputedso that the DFT can be computedas a vector-matrix multiplication.

• Later we will exploit the symmetryproperties of the exponential tospeed up the computation (e.g., fft()).

The Discrete-Time Fourier Transform

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EE 3512: Lecture 17, Slide 3

Computation of the DFT

• Given the signal:

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3sin()sin(2)

2sin(2

)2

3cos()cos(2)

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EE 3512: Lecture 17, Slide 4

Symmetry

• The magnitude and phase functions are even and odd respectively.

• The DFT also has “circular” symmetry:

• When N is even, |Xk| is symmetric about N/2.

• The phase, Xk, has odd symmetry about N/2.

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EE 3512: Lecture 17, Slide 5

Inverse DFT

• The inverse transform follows from the DT Fourier Series:

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EE 3512: Lecture 17, Slide 6

Computation of the Inverse DFT

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EE 3512: Lecture 17, Slide 7

Relationship to the DTFT

• The DFT and the DFT are related by:

• If we define a pulse as:

• The DFT is simply a sampling of theDTFT at equispaced points along thefrequency axis.

• As N increases, the sampling becomesfiner. Note that this is true even whenq is constant increasing N is a way ofinterpolating the spectrum.

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• q = 5, N = 88

• q = 5

EE 3512: Lecture 17, Slide 8

DFT of Truncated Signals• What if the signal is not time-limited?

We can think of limiting the sum toN points as a truncation of the signal:

• What are the implications of this in the frequency domain?(Hint: convolution)

• Popular Windows: Rectangular:

Generalized Hanning:

Triangular:

otherwise

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otherwise

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• Rectangular

• Generalized Hanning

• Triangular

EE 3512: Lecture 17, Slide 9

Impact on Spectral Estimation• The spectrum of a windowed

sinewave is the convolution of two impulse functions with the frequency response of the window.

• For two closely spaced sinewaves, there is “leakage” between each sinewave’s spectrum.

• The impact of this leakage can be mitigated by using a window function with a narrower main lobe.

• For example, consider the spectrum of three sinewaves computed using a rectangular and a Hamming window.

• We see that for the same number of points, the spectrum produced by te Hamming window separates the sinewaves.

• What is the computational cost?

EE 3512: Lecture 17, Slide 10

Summary• Introduced the Discrete Fourier Transform as a truncated version of the

Discrete-Time Fourier Transform.

• Demonstrated both the forward and inverse transforms.

• Explored the relationship to the DTFT.

• Compared the spectrum of a pulse.

• Discussed the effects of truncation on the spectrum.

• Introduced the concept of time domain windowing and discussed the impact of windows in the frequency domain.