This document provides an overview of frequency domain representation of signals and systems. It defines key concepts such as the Fourier transform, which converts a signal from the time domain to the frequency domain. The frequency spectrum shows the distribution of frequencies within a signal. Periodic signals can be represented using Fourier series, while aperiodic signals use the Fourier transform. Properties of the Fourier transform such as linearity, time shifting, and the convolution theorem are also covered.

1. Frequency Domain
Representation of Signals and
Systems
Prof. Satheesh Monikandan.B
INDIAN NAVAL ACADEMY (INDIAN NAVY)
EZHIMALA
sathy24@gmail.com
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,
Aej(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 radiansϕ
or degrees)

Both A and f are real and non-negative.

Complex exponential can also written as, Aej(ωt+ )ϕ

From Euler’s relation, ejωt
=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) = XR
(f) + jXI
(f) = |X(f)|ejθ(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 x1
(t) ← → X⎯ 1
(f) and x2
(t) ← → X⎯ 2
(f)
 Then, for all constants a1
and a2
, we have
a1
x1
(t) + a2
x2
(t) ← → a⎯ 1
X1
(f) + a2
X2
(f)

Time Scaling

23. Properties of Fourier Transform

Time shift
 If x(t) ← → X(f) then, x(t−t⎯ 0
) ← → e⎯ -2πft
0X(f)
 If t0
is positive, then x(t−t0
) is a delayed version of x(t).
 If t0
is negative, then x(t−t0
) 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−t0
) 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.