3.Frequency Domain Representation of Signals and Systems

Fourier Series , Properties, CTFT and Properties

1. Frequency Domain
Representation of Signals and
Systems
Prof. Satheesh Monikandan.B
INDIAN NAVAL ACADEMY (INDIAN NAVY)
EZHIMALA
[email protected]
101 INAC-AT19

2. Syllabus Contents
•Introduction to Signals and Systems
•Time-domain Analysis of LTI Systems
•Frequency-domain Representations of Signals and
Systems
•Sampling
•Hilbert Transform
•Laplace Transform

3. Frequency Domain Representation of
Signals

Refers to the analysis of mathematical functions or
signals with respect to frequency, rather than time.

"Spectrum" of frequency components is the
frequency-domain representation of the signal.

A signal can be converted between the time and
frequency domains with a pair of mathematical
operators called a transform.

Fourier Transform (FT) converts the time function
into a sum of sine waves of different frequencies,
each of which represents a frequency component.

Inverse Fourier transform converts the frequency
(spectral) domain function back to a time function.

4. Frequency Spectrum

Distribution of the amplitudes and phases of each
frequency component against frequency.

Frequency domain analysis is mostly used to signals
or functions that are periodic over time.

5. Periodic Signals and Fourier Series

A signal x(t) is said to be periodic if, x(t) = x(t+T) for
all t and some T.

A CT signal x(t) is said to be periodic if there is a
positive non-zero value of T.

6. Fourier Analysis

The basic building block of Fourier analysis is the
complex exponential, namely,
Ae
j(2πft+ )
ϕ
or Aexp[j(2πft+ )]
ϕ
where, A : Amplitude (in Volts or Amperes)
f : Cyclical frequency (in Hz)
: Phase angle at t = 0 (either in radians
ϕ
or degrees)

Both A and f are real and non-negative.

Complex exponential can also written as, Ae
j(ωt+ )
ϕ

From Euler’s relation, e
jωt
=cosωt+jsinωt

7. Fourier Analysis

Fundamental Frequency - the lowest frequency which is
produced by the oscillation or the first harmonic, i.e., the
frequency that the time domain repeats itself, is called the
fundamental frequency.

Fourier series decomposes x(t) into DC, fundamental and
its various higher harmonics.

8. Fourier Series Analysis

Any periodic signal can be classified into harmonically related
sinusoids or complex exponential, provided it satisfies the
Dirichlet's Conditions.

Fourier series represents a periodic signal as an infinite sum of sine
wave components.

Fourier series is for real-valued functions, and using the sine and
cosine functions as the basis set for the decomposition.

Fourier series can be used only for periodic functions, or for
functions on a bounded (compact) interval.

Fourier series make use of the orthogonality relationships of the
sine and cosine functions.

It allows us to extract the frequency components of a signal.

9. Fourier Coefficients

11. Fourier Coefficients

Fourier coefficients are real but could be bipolar (+ve/–ve).

Representation of a periodic function in terms of Fourier
series involves, in general, an infinite summation.

12. Convergence of Fourier Series

Dirichlet conditions that guarantee convergence.
1. The given function is absolutely integrable over any
period (ie, finite).
2. The function has only a finite number of maxima and
minima over any period T.
3.There are only finite number of finite discontinuities over
any period.

Periodic signals do not satisfy one or more of the above
conditions.

Dirichlet conditions are sufficient but not necessary.
Therefore, some functions may voilate some of the
Dirichet conditions.

13. Convergence of Fourier Series

Convergence refers to two or more things coming
together, joining together or evolving into one.

14. Applications of Fourier Series

Many waveforms consist of energy at a fundamental
frequency and also at harmonic frequencies (multiples of
the fundamental).

The relative proportions of energy in the fundamental and
the harmonics determines the shape of the wave.

Set of complex exponentials form a basis for the space of
T-periodic continuous time functions.

Complex exponentials are eigenfunctions of LTI systems.

If the input to an LTI system is represented as a linear
combination of complex exponentials, then the output can
also be represented as a linear combination of the same
complex exponential signals.

15. Parseval’s (Power) Theorem

Implies that the total average power in x(t) is the
superposition of the average powers of the complex
exponentials present in it.

If x(t) is even, then the coeffieients are purely real and
even.

If x(t) is odd, then the coefficients are purely imaginary
and odd.

16. Aperiodic Signals and Fourier Transform

Aperiodic (nonperiodic) signals can be of finite or infinite
duration.

An aperiodic signal with 0 < E < ∞ is said to be an energy
signal.

Aperiodic signals also can be represented in the
frequency domain.

x(t) can be expressed as a sum over a discrete set of
frequencies (IFT).

If we sum a large number of complex exponentials, the
resulting signal should be a very good approximation to
x(t).

17. Fourier Transform

Forward Fourier transform (FT) relation, X(f)=F[x(t)]

Inverse FT, x(t)=F
-1
[X(f)]

Therefore, x(t) ← → X(f)
⎯

X(f) is, in general, a complex quantity.
Therefore, X(f) = X
R
(f) + jX
I
(f) = |X(f)|e
jθ(f)

Information in X(f) is usually displayed by means of two
plots:
(a) X(f) vs. f , known as magnitude spectrum
(b) θ(f) vs. f , known as the phase spectrum.

18. Fourier Transform

FT is in general complex.

Its magnitude is called the magnitude spectrum and its
phase is called the phase spectrum.

The square of the magnitude spectrum is the energy
spectrum and shows how the energy of the signal is
distributed over the frequency domain; the total energy of
the signal is.

19. Fourier Transform

Phase spectrum shows the phase shifts between signals
with different frequencies.

Phase reflects the delay (relationship) for each of the
frequency components.

For a single frequency the phase helps to determine
causality or tracking the path of the signal.

In the harmonic analysis, while the amplitude tells you
how strong is a harmonic in a signal, the phase tells where
this harmonic lies in time.

Eg: Sirene of an rescue car, Magnetic tape recording,
Auditory system

The phase determines where the signal energy will be
localized in time.

20. Fourier Transform

21. Properties of Fourier Transform

Linearity

Time Scaling

Time shift

Frequency Shift / Modulation theorem

Duality

Conjugate functions

Multiplication in the time domain

Multiplication of Fourier transforms / Convolution theorem

Differentiation in the time domain

Differentiation in the frequency domain

Integration in time domain

Rayleigh’s energy theorem

22. Properties of Fourier Transform

Linearity
Let x
1
(t) ← → X
⎯
1
(f) and x
2
(t) ← → X
⎯
2
(f)
Then, for all constants a
1
and a
2
, we have
a
1
x
1
(t) + a
2
x
2
(t) ← → a
⎯
1
X
1
(f) + a
2
X
2
(f)

Time Scaling

23. Properties of Fourier Transform

Time shift
If x(t) ← → X(f) then, x(t−t
⎯
0
) ← → e
⎯
-2πft
0 X(f)
If t
0
is positive, then x(t−t
0
) is a delayed version of x(t).
If t
0
is negative, then x(t−t
0
) is an advanced version of x(t) .

Time shifting will result in the multiplication of X(f) by a
linear phase factor.
x(t) and x(t−t
0
) have the same magnitude spectrum.

24. Properties of Fourier Transform

Frequency Shift / Modulation theorem

25. Properties of Fourier Transform

Multiplication in the time domain

26. Properties of Fourier Transform

Multiplication of Fourier transforms / Convolution theorem

Convolution is a mathematical way of combining two
signals to form a third signal.

The spectrum of the convolution two signals equals the
multiplication of the spectra of both signals.

Under suitable conditions, the FT of a convolution of two
signals is the pointwise product of their Fts.

Convolving in one domain corresponds to elementwise
multiplication in the other domain.

27. Properties of Fourier Transform

Multiplication of Fourier transforms / Convolution theorem

28. Properties of Fourier Transform

Differentiation in the time / frequency domains

29. Properties of Fourier Transform

Rayleigh’s energy theorem

30. Parseval’s Relation

The sum (or integral) of the square of a function is equal to
the sum (or integral) of the square of its transform.

The integral of the squared magnitude of a function is
known as the energy of the function.

The time and frequency domains are equivalent
representations of the same signal, they must have the
same energy.

31. Time-Bandwidth Product

Effective duration and effective bandwidth are only useful
for very specific signals, namely for (real-valued) low-pass
signals that are even and centered around t=0.

This implies that their FT is also real-valued, even and
centered around ω=0, and, consequently, the same
definition of width can be used.

Two different shaped waveforms to have the same time-
bandwidth product due to the duality property of FT.