Digital Signal Processing: Discrete Fourier Transform (DFT)

Name: Digital Signal Processing: Discrete Fourier Transform (DFT)
Rating: 4.5 (17 reviews)

DFT, Properties of DFT

Created byKoti Reddy

Last updated 1/2021

English

What you'll learn

Discrete Fourier Transform
Properties of DFT
Inverse Discrete Fourier Transform
Circular convolution

Course content

1 section • 14 lectures • 1h 59m total length

Introduction12:40
Introduce the discrete Fourier transform (DFT) and the role of sampling in converting continuous signals to discrete data, highlighting sample count and timing implications.
Procedure to find DFT18:58
Learn the procedure to find the discrete Fourier transform (DFT), demonstrated with omega-based calculations and a three-by-three matrix, to illustrate expanding the DFT step by step.
Periodicity property, DFT of a real valued signal9:29
Explore the periodicity property in the DFT of real-valued signals, including conjugate relationships and practical examples.
Example problems7:16
Work through example problems in the discrete Fourier transform, applying DFT properties to analyze the signal and compute key results.
Circular convolution, Parseval's Theorem20:32
Explore circular convolution and Parseval's theorem within the discrete Fourier transform framework. Analyze how convolution interacts with samples, matrix multiplication, and signal properties.
Time expansion property5:00
Examine the time expansion property of the discrete Fourier transform, showing how a signal's samples expand in time and how this affects the dft representation.
Performing Linear convolution using Circular convolution8:25
Learn to perform linear convolution using circular convolution by aligning two signals with the same number of samples and applying multiplication and addition across them.
Circular time reversal8:44
examine circular time reversal in the discrete Fourier transform, showing how reversing a time sequence affects the frequency-domain representation and sample behavior.
Expansion of x(n) , Expansion of X(k)9:41
Expand x(n) by two, turning each sample into two, and analyze the corresponding expansion of X(k) in the DFT, including two-point DFT steps and resulting DFT values.
Example Problem5:37
Solve an example problem illustrating the discrete Fourier transform, processing sample values to compute DFT components and interpret the results within digital signal processing.
Example Problem4:34
Examine a six-point discrete Fourier transform example problem in digital signal processing, analyzing frequency indices and DFT values to determine the correct result.
Example Problem2:22
Explore a quick shortcut for solving a discrete Fourier transform example by leveraging impulse properties, avoiding the standard formula, and understanding how the DFT treats impulses.
Example Problem2:30
This example clarifies how the discrete Fourier transform relates time-domain convolution to frequency-domain multiplication, illustrating circular convolution of ABC with ABC within the DFT framework.
Example Problem3:38
Explore a four-point discrete Fourier transform through a worked example, showing how multiplying the dft matrix by data identifies the nonzero frequency component, emphasizing exactness.

Requirements

Basics of Signals and Systems

Description

The Discrete Fourier Transform (DFT) is one of the most important tools in Digital Signal Processing. It enables us to find the spectrum of a finite duration sequence.

Discrete Fourier Transform (DFT) is an algorithm to implement Discrete Time Fourier Transform (DTFT) on computers for signal processing by sampling at equally spaced frequency points of one cycle of DTFT. Unlike DTFT (DTFT is continuous), the output of DFT is discrete and hence can be implemented on a computer.

DTFT is a continuous function of frequency, where as DFT is a discrete function of frequency.

Computing the DFT is equivalent to solving a set of linear equations.

In direct DFT method, the computational part is too long.

The Discrete Fourier Transform (DFT) can be computed efficiently using a Fast Fourier Transform (FFT).

Fast Fourier Transform (FFT) is an algorithm to reduce number of complex additions and complex multiplications while calculating DFT.

In this course, I have explained about Discrete Fourier Transform ( DFT) and Inverse DFT (IDFT).

The topics are

Relationship between DFT and DTFT

Finding DFT for a given x(n)

Properties of DFT

Linear convolution

Circular convolution or Periodic convolution

Difference between circular convolution and linear convolution

Performing linear convolution using circular convolution

Many example problems are also explained on all concepts.

After the completion of this course, definitely you are able solve problems on your own.

Who this course is for:

Engineering students