Discrete Fourier Series | Discrete Fourier Transform | Discrete Time Fourier Transform

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Discrete Fourier Series | Discrete Fourier Transform | Discrete Time
Fourier Transform
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DISCRETE FOURIER SERIES
DISCRETE FOURIER TRANSFORM
DISCRETE TIME FOURIER TRANSFORM
Meer Zafarullah Noohani K-F16ES-33
Kaleem Ullah Magsi K-F16ES-57
Ms. Sumera Hashim K-F16ES-53

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presentations are intended only for the learning purpose.
8/2/2020 2

DISCRETE FOURIER SERIES
• The discrete Fourier series is a periodic function
• Fourier series represent either:
(a) A periodic function, or
(b) A function that is defined only over a finite-length interval; the values
produced by the Fourier series outside the finite interval are irrelevant.
4

• When the function being represented is discrete, that set is called a discrete
Fourier transform (DFT).
• The term discrete Fourier series (DFS) is intended for use instead
of DFT when the original function is periodic, defined over an infinite interval.
• DFS is a frequency analysis tool for periodic infinite-duration discrete-time
signals.
5

6
The discrete Fourier
series (DFS) is called
“the Fourier transform for
periodic sequences,” in that it
plays the same role for them
that the Fourier transform
plays for nonperiodic
(ordinary) sequences.
These are discrete signals that
repeat themselves in a
periodic fashion from negative
to positive infinity. This class
of Fourier transform is
sometimes called the
discrete Fourier series, but is
most often called the
discrete Fourier transform.

Periodicity in Discrete Fourier Series
7

APPLICATIONS OF DFS
Signal Processing.
The best application of
Fourier analysis.
Approximation Theory.
Used to write a function
as a trigonometric
polynomial.

Control Theory.
Useful to find out the dynamics of the solution.
Partial Differential equation.
We use it to solve higher order partial differential equations
by the method of separation of variables.
The Fourier series of functions in the differential equation
often gives some prediction about the behavior of the
solution of differential equation. 14

Computation of power spectra.
Cross spectra.
Coherence.

One dimensional heat equation and wave equation
Advanced noise cancellation
Cell phone network technology

• Original time function can be uniquely recovered from it.
• Fourier methods give us a set of powerful tools for representing
any periodic function as a sum of sines and cosines.
• The main advantage of Fourier analysis is that very little
information is lost from the signal during the transformation.
17
ADVANTAGES

18
FREQUENCY EXTRACTION
SPECTRUM
ANALYSIS
BEAMFORMING
AUDIO
PROCESSING
ADVANTAGES

• All periodic functions can't be expressed in terms of Fourier
series, but only those which follow Dirichlet conditions.
• It can be used only for periodic inputs and thus not applicable
for aperiodic one. (Fourier Transform)
• It cannot be used for unstable or even marginally stable
systems. (Laplace transform)
19
DISADVANTAGES

INTRODUCTION
• The DFT is one of the most powerful tools in digital signal processing which
enables us to find the spectrum of a finite-duration signal.
• There are many circumstances in which we need to determine the frequency
content of a time-domain signal.
• For example, we may have to analyze the spectrum of the output of an LC
oscillator to see how much noise is present in the produced sine wave. This
can be achieved by the discrete Fourier transform (DFT).

• Basically, computing the DFT is equivalent to solving a set of linear equations.
• The DFT provides a representation of the finite-duration sequence using a
periodic sequence, where one period of this periodic sequence is the same as the
finite-duration sequence.
• As a result, we can use the discrete-time Fourier series to derive the DFT
equations.
• The algorithms for the efficient computation of the DFT are collectively called fast
Fourier transforms (FFTs).
22

WHAT IS DFT?
▪ Assume that x(t)x(t), shown in
Figure 1, is the continuous-time
signal that we need to analyze.
24
Figure 1

▪ Assume that the finite-duration sequence that we need to analyze is as shown
in Figure 5 (a). To calculate the N-point DFT, we need to make a periodic
signal, p(n)p(n), from x(n)x(n) with period NN, as shown in Figure 5(b).
▪ Considering the fact that p(n)=x(n)p(n)=x(n) for n=0,1,…,N−1n=0,1,…,N−1, we
obtain the discrete-time Fourier series of this periodic signal
▪ Equation 1
25

• Where N denotes the period of the signal. The time-domain signal can be
obtained as follows:
• Equation 2
26

▪ Multiplying the coefficients given by Equation 6 by NN, we obtain the DFT
coefficients, X(k)X(k), as follows:
▪ Equation 3 →
▪ The inverse DFT will be
▪ Equation 4 →
27

Discrete Fourier Series | Discrete Fourier Transform | Discrete Time Fourier Transform

• To what degree does the DFT approximate the Fourier
transform of the function underlying the data? Clearly
the DFT is only an approximation since it provides only
for a finite set of frequencies. But how correct are these
discrete values themselves? There are two main types of
DFT errors:
• 1. Aliasing
• 2. Leakage
32
ERRORS IN DFT

ALIASING
• If the initial samples are not sufficiently closely spaced to
represent high-frequency components present in the
underlying function, then the DFT values will be corrupted
by aliasing
33

LEAKAGE
• The continuous Fourier Transform requires the integration to be
performed over the interval – P to FP or over an integer number of cycles
of the waveform. If we attempt to complete the DFT over a non-integer
number of cycles of the input signal, then we might expect the transform
to be corrupted in some way
34

Advantages
The main advantage of discrete Fourier Transform analysis
is that very little information is lost from the signal during
the transformation.
The DFT can find a system's frequency response from the
system's impulse response, and vice versa.
The DFT is especially useful for efficiently representing
signals that are comprised of a few frequency components.
Like ECG recorded signal etc.
35
ADVATAGES

Applications
Image filtering Seismograms-
Frequency content
Eigenmodes of the
Earth
Seismo-weather
with FFTS

IMAGE APPLICATIONS, WHICH INCLUDES IMAGE ANALYSIS, IMAGE
FILTERING, IMAGE RECONSTRUCTION AND IMAGE COMPRESSION.
INFRA-RED SPECTROSCOPY
FT-RAMAN SPECTROSCOPY
MEDICAL MRI IMAGING

DISCRETE TIME FOURIER TRANSFORM
38

DEFINITION OF DTFT
• In mathematics, the discrete-time Fourier transform is a form of Fourier
analysis that is applicable to a sequence of values. The DTFT is often
used to analyze samples of a continuous function.
39

OR
• The DTFT is a frequency-domain representation for a wide range of both finite
and infinite-length discrete-time signals x[n].
• The DTFT is denoted as X(ejωˆ), which shows that the frequency dependence
always includes the complex exponential function ejωˆ .
• The operation of taking the Fourier transform of a signal will become a
common tool for analyzing signals and systems in the frequency domain.
40

• The application of the DTFT is usually called Fourier analysis, or spectrum
analysis or “going into the Fourier domain or frequency domain.”
• Thus, the words spectrum, Fourier, and frequency-domain representation
become equivalent, even though each one retains its own distinct character.
41

DISCRETE-TIME SIGNALS IN FREQUENCY DOMAIN
• For continuous-time signals, we can use Fourier series and Fourier transform
to study them in frequency domain.
• With the use of sampled version of a continuous-time signal we can
obtain the discrete-time Fourier transform (DTFT) or Fourier transform of
discrete-time signals as follows.
42

We start with studying the sampled signal produced by multiplying by
the impulse train
43

APPLICATIONS
Orthogonal Frequency
Division Multiplex (OFDM)
Image compression to
MPEG format
Digital filters

Thank You
50
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