2. Non-periodic Signals: Fourier-Transform
Representations
No restrictions on the period of the sinusoids used to represent
non-periodic signal.
Frequencies can take a continuum of values.
For CT non periodic signal the range is from −∞ to ∞
For DT non periodic signal the range is from −ߨ to ߨ
CTFT
ݔ ݐ =
1
2ߨ
න ܺ(݆߱)݁ఠ௧݀߱
ஶ
ିஶ
(1)
DTFT
ݔ ݊ =
1
2ߨ
න ܺ(݆Ω)݁Ω݀Ω
గ
ିగ
(2)
Prof: Sarun Soman, MIT, Manipal 2
3. Continuous Time Non-periodic Signals: The
Fourier Transform
CTFT is used to represent a continuous time non-periodic signal
as a superposition of complex sinusoids.
ݔ ݐ =
1
2ߨ
න ܺ(݆߱)݁ఠ௧݀߱
ஶ
ିஶ
Where
ܺ ݆߱ = න ݁)ݐ(ݔିఠ௧݀ݐ
ஶ
ିஶ
ܺ ݆߱ is the frequency domain representation of )ݐ(ݔ
The weight on each sinusoid is
ఠ
ଶగ
Prof: Sarun Soman, MIT, Manipal 3
4. Continuous Time Non-periodic Signals: The
Fourier Transform
CTFT is used to analyze the characteristics of CT systems and the
interaction b/w CT signals and systems.
Eq(1) and (2) may not converge for all functions of x(t)
Dirichlet conditions for non periodic signal
x(t) is absolutely integrable
න )ݐ(ݔ ݀ݐ < ∞
ஶ
ିஶ
x(t) has a finite number of maxima, minima and discontinuities in any
finite interval.
The size of each discontinuity is finite
Eg. Unit step function is not absolutely integrable
Prof: Sarun Soman, MIT, Manipal 4
12. Example
=
sin ܹݐ
ߨݐ
)ݐ(ݔ =
ܹ
ߨ
sin ܹݐ
ܹݐ
, ݐ ≠ 0
For ݐ = 0
lim
௧→
sin ܹݐ
ߨݐ
ݔ ݐ =
ܹ
ߨ
Zero crossing points
ܹݐ = ±݉ߨ, ݉ = ±1,2,3 … .
ݐ = ±
݉ߨ
ܹ
Prof: Sarun Soman, MIT, Manipal 12
13. Properties of Fourier Transform
Linearity
Linearity property is the basis of the partial fraction method for
determining inverse FT.
Eg.
Find )ݐ(ݔ
ܺ ݆߱ =
−݆߱
(݆߱)ଶ+3݆߱ + 2
ܽݔ ݐ + ܾݔ ݐ ܽܺ ݆߱ + ܾܻ(݆߱)
Prof: Sarun Soman, MIT, Manipal 13
14. Example
=
ܿଵ
݆߱ + 1
+
ܿଶ
݆߱ + 2
ܿଵ = 1, ܿଶ = −2
ܺ ݆߱ =
1
݆߱ + 1
−
2
݆߱ + 2
Using the transformation table
1
1
݆߱ + 1
+ −2
1
݆߱ + 2
↔ 1 ݁ି௧ݑ ݐ
+ (−2)݁ିଶ௧)ݐ(ݑ
ݔ ݐ = ݁ି௧ݑ ݐ − 2݁ିଶ௧)ݐ(ݑ
Symmetry Property: Real and
Imaginary Signals.
If )ݐ(ݔ is real and even
ܺ(݆߱) is real
If )ݐ(ݔ is real and odd
ܺ(݆߱) is imaginary
Time Shift properties
ݐ(ݔ − ݐ) ↔ ݁ିఠబ௧ܺ(݆߱)
• Shift in time domain leaves the
magnitude spectrum unchanged
• Introduces a phase shift that is
linear function of
frequency(݁ିఠబ௧).
݁ି௧)ݐ(ݑ ↔
1
݆߱ + ܽ
Prof: Sarun Soman, MIT, Manipal 14
15. Properties of Fourier Transform
Differentiation Property
Differentiation in time
݀
݀ݐ
)ݐ(ݔ ↔ ݆߱ܺ(݆߱)
• Differentiation in time domain
corresponds to multiplying by j߱
in frequency domain.
• This operation accentuates high
frequency components.
Eg.
݁ି௧)ݐ(ݑ ↔
1
݆߱ + ܽ
݀
݀ݐ
݁ି௧)ݐ(ݑ ↔ (݆߱)
1
݆߱ + ܽ
Differentiation in Frequency
−݆ݐ )ݐ(ݔ ↔
݀
݀߱
ܺ(݆߱)
Eg.
Use differentiation property to find
FT of ݔ ݐ = ݁ݐି௧)ݐ(ݑ
Ans:
Using differentiation property
−݆ݐ )ݐ(ݔ ↔
݀
݀߱
ܺ(݆߱)
ݐ )ݐ(ݔ ↔
1
−݆
݀
݀߱
ܺ(݆߱)
݁ݐି௧ ↔ ݆
݀
݀߱
1
݆߱ + ܽ
Prof: Sarun Soman, MIT, Manipal 15
16. Properties of Fourier Transform
݁ݐି௧
↔
1
݆߱ + ܽ ଶ
Integration
න ݔ ߬ ݀߬ =
1
݆߱
ܺ ݆߱ + ߨܺ(݆0)ߜ(߱)
௧
ିஶ
• De emphasizing high frequency
components.
Eg.
FT of unit step using integration
property
Ans:
ݑ ݐ = න ߜ ߬ ݀߬
௧
ିஶ
ߜ()ݐ ↔ 1
Using integration property
න ߜ ߬ ݀߬
௧
ିஶ
↔
1
݆߱
1 + ߨߜ ߱
Convolution property
ݔ ݐ ∗ ݄()ݐ ↔ ܺ ݆߱ )݆߱(ܪ
Eg.
Let the input to a system with impulse
response ݄ ݐ = 2݁ିଶ௧
)ݐ(ݑ be
ݔ ݐ = 3݁ି௧
ݑ ݐ .
Prof: Sarun Soman, MIT, Manipal 16
18. Properties of Fourier Transform
• Slope of the linear phase term is
equal to the time shift (ݐ).
Eg.
ݔ ݐ = ݁ି௧ାଶ
ݐ(ݑ − 2)
Ans:
݁ି௧
ݑ ݐ ↔
1
݆߱ + 1
݁ି௧ାଶ
ݐ(ݑ − 2) ↔ ݁ିఠ(ଶ)
1
݆߱ + 1
Frequency Shift Properties
݁ఊ௧
)ݐ(ݔ ↔ ܺ(݆(߱ − ߛ))
• A frequency shift corresponds to
multiplication in time domain by a
complex sinusoid whose frequency
is equal to the shift.
Eg.
ݔ ݐ ↔
2
߱
sin(߱ߨ)
݁ଵ௧)ݐ(ݔ
↔
2
߱ − 10
sin(ߨ(߱ − 10))
Scaling Property
)ݐ(ݔ ↔ ܺ(݆߱)
)ݐܽ(ݔ ↔
1
ܽ
ܺ ݆
߱
ܽ
Scaling the signal in time domain
introduces inverse scaling in
frequency domain representation &
an amplitude scaling.
Prof: Sarun Soman, MIT, Manipal 18
19. Properties of Fourier Transform
Parseval’s Theorem
Parseval’s theorem states that energy or power in time domain
representation is equal to the energy or power in frequency
domain.
න )ݐ(ݔ ଶ݀ݐ
ஶ
ିஶ
=
1
2ߨ
න ܺ(݆߱) ଶ݀߱
ஶ
ିஶ
Duality property
There is a consistent symmetry b/w the time and Frequency
domain representation of signals.
A rectangular pulse in either time or frequency domain
corresponds to a sinc function in either frequency or time.
Prof: Sarun Soman, MIT, Manipal 19
20. Properties of Fourier Transform
We may interchange time and frequency
This interchangeability property is termed duality.
Prof: Sarun Soman, MIT, Manipal 20
21. Properties of Fourier Transform
݂()ݐ
ி்
ܨ ݆߱
)ݐ݆(ܨ
ி்
2ߨ݂(−߱)
Using duality property find the
duality property of ‘1’
Ans:
ߜ()ݐ
ி்
1
1
ி்
2ߨߜ −߱
Find the FT of ݔ ݐ =
ଵ
ଵା௧
Ans:
݁ି௧ݑ ݐ
ி் 1
݆߱ + 1
Replace ߱ by ݐ
1
݆ݐ + 1
Prof: Sarun Soman, MIT, Manipal 21
23. Discrete Time Non-periodic Signals: The
Discrete Time Fourier Transform
DTFT is used to represent a discrete-time -periodic signal as a
superposition of complex sinusoids.
DTFT would involve a continuum of frequencies on the
interval−ߨ < Ω < ߨ
ݔ ݊ =
1
2ߨ
න ܺ(݁Ω
)݁Ω
݀
గ
ିగ
Ω
Where
ܺ ݁Ω = ݁]݊[ݔିΩ
ஶ
ୀିஶ
ܺ ݁Ω is termed as the frequency domain representation of
]݊[ݔ
Prof: Sarun Soman, MIT, Manipal 23
24. Example
Find the DTFT of the exponential
sequence ݔ ݊ =
ଵ
ସ
݊[ݑ + 4]
Ans:
ܺ ݁Ω = ݁]݊[ݔିΩ
ஶ
ୀିஶ
=
1
4
݁ିΩ
ஶ
ୀିସ
Let ݊ + 4 = ݈
=
1
4
ିସ
݁ିΩ(ିସ)
ஶ
ୀ
=
1
4
ିସ
݁Ωସ
1
4
݁ିΩ
ஶ
ୀ
= 256݁ସΩ
1
1 −
1
4
݁ିΩ
Evaluate the DTFT of signal x[n]
shown in Fig. Find the expression for
magnitude and phase spectra.
0 1 2
3
-1-2-3
n
]݊[ݔ
1
-1
Prof: Sarun Soman, MIT, Manipal 24
31. z transform
• DTFT- complex sinusoidal representation of a DT signal
• ݖ transform – Representation in terms of complex exponential
signals.
• ݖ transform is the discrete time counterpart to Laplace
transform
Why ݖ transform?
• More general classification of DT signal.
• A broader characterization of DT LTI systems & its interaction
with signals.
Prof: Sarun Soman, MIT, Manipal 31
32. Z transform
Eg.
DTFT exists only if impulse response is absolutely summable.
DTFT exists only for stable LTI systems.
ݖ transform of the impulse response exists for unstable LTI
systems and signals.
ݖ transform of the impulse response is the transfer function of
the system.
ݖ = ݁ݎΩ
ݎ − ݉ܽ݃݊݅,݁݀ݑݐ Ω − ݈ܽ݊݃݁
ݔ ݊ = ݖ complex exponential signal.
Prof: Sarun Soman, MIT, Manipal 32
33. Z transform
ݔ ݊ = ݎ cos Ω݊ + ݆ݎ sin Ω݊
If ݎ = 1, ]݊[ݔ is a complex sinusoid.
Applying ]݊[ݔ to an LTI system
ݕ ݊ = ݄ ݊ ∗ ]݊[ݔ
= ݄ ݇ ݊[ݔ − ݇]
ஶ
ୀିஶ
ݔ ݊ = ݖ
ݕ ݊ = ݄[݇]ݖି
ஶ
ୀିஶ
Prof: Sarun Soman, MIT, Manipal 33
34. z transform
= ݖ
݄[݇]ݖି
ஶ
ୀିஶ
Transfer function
ܪ ݖ = ݄[݇]ݖି
ஶ
ୀିஶ
ݖ transform of ]݊[ݔ
ܺ ݖ = ݖ]݊[ݔି
ஶ
ୀିஶ
(1)
Convergence
• ݖ transform exist when eqn(1)
converges.
• Necessary condition is absolute
summability.
ݖ]݊[ݔି
ஶ
ୀିஶ
< ∞ (2)
ݖ = ݁ݎΩ
ݖି = ݎି
Equation (2) can be written as
ݎ]݊[ݔି
ஶ
ୀିஶ
< ∞
Prof: Sarun Soman, MIT, Manipal 34
35. z transform
• The range ′′ݎ for which eq(2) converges is termed as Region of
Convergence(ROC)
• ݎ]݊[ݔି is absolutely summable even though ]݊[ݔ is not.
• Ability to work with signals that doesn't have a DTFT is a
significant advantage offered by the ݖ transform.
Z-plane.
Prof: Sarun Soman, MIT, Manipal 35
36. transform
ࢠ transform of a causal exponential
signal
Determine the ݖ transform of the
signal ݔ ݊ = ߙ
.]݊[ݑ Depict the
ROC and the location of poles and
zeros of ܺ()ݖ in the ݖ plane.
Ans:
ܺ ݖ = ݖ]݊[ݔି
ஶ
ୀିஶ
ܺ ݖ = ߙݖ]݊[ݑି
ஶ
ୀିஶ
=
ߙ
ݖ
ஶ
ୀ
The sum converges only if
ߙ
ݖ
< 1
ݖ > ߙ
ܺ ݖ =
1
1 − ߙݖିଵ
, ݖ > ߙ
ܺ()ݖin pole-zero form
=
ݖ
ݖ − ߙ
, ݖ > ߙ
Pole zero plot and ROC
Prof: Sarun Soman, MIT, Manipal 36
37. ݖ transform
ࢠ transform of non-causal
exponential signal
Determine the ݖ transform of the
signal ݕ ݊ = −ߙݑ −݊ − 1 .Depict
the ROC and the locations of poles
and zeros of ܺ ݖ in the ݖ plane.
Ans:
ܻ ݖ = ݖ]݊[ݕି
ஶ
ୀିஶ
= − ߙ
ିଵ
ିஶ
ݖି
Let ݇ = −݊
ܻ ݖ = −
ݖ
ߙ
ஶ
ୀଵ
= −
ݖ
ߙ
ஶ
ୀ
− 1
= 1 −
ݖ
ߙ
ஶ
ୀ
The sum converges, provided
௭
ఈ
< 1
ݖ < ߙ
= 1 −
1
1 − ߙݖିଵ
, ݖ < ߙ
Prof: Sarun Soman, MIT, Manipal 37
38. transform
=
1 − ߙݖିଵ − 1
1 − ߙݖିଵ
=
−ߙݖିଵ
1 − ߙݖିଵ
= −
ݖ
ߙ − ݖ
=
ݖ
ݖ − ߙ
, ݖ < ߙ
ROC plot
ݖ transform is same but ROC is
different
z transform of a two sided signal
Determine the z-transform of
ݔ ݊ = −ݑ −݊ − 1 +
ଵ
ଶ
.]݊[ݑ
Depict the ROC and the locations of
poles and zeros of ܺ()ݖ in the plane.
ܺ ݖ =
1
2
ݖ]݊[ݑି
ஶ
ୀିஶ
− ݊−[ݑ
− 1]ݖି
=
1
2ݖ
−
1
ݖ
ିଵ
ୀିஶ
ஶ
ୀ
=
1
2ݖ
+ 1 − ݖ
ஶ
ୀ
ஶ
ୀ
Both the sum converges when
ݖ >
1
2
ܽ݊݀ ݖ < 1
Prof: Sarun Soman, MIT, Manipal 38