Fourier Transform Representation from
Fourier Series
► Substiute eq(2) in eq(1)………
Converting parameters periodic
Fourier
signal to non-periodic signal
Transform
(F.S to F.T)
Inverse-Fourier Transform
Fourier Series Representation from
Fourier Transform Representation
Fourier Transform of Gate Function
Continue
Sampling
Function
Fourier Transform of Impulse Function
Fourier Transform of DC Signal
Approximation
Lets suppose we take approximation to Gate function
Fourier Transform of Cosine Function
Applying Sampling
Function
Fourier Transform Example:
Magnitude
response
Magnitude Response
Phase Response
Phase Response
Prove Fourier Transform Properties
Fourier Transform of given Energy
Signals
Inverse Fourier Transform
Example: Find Inverse Fourier Transform of
following Signal.
Parseval’s Theorem & Parseval’s Identity of
Fourier Transform
Parseval’s theorem states that the energy of signal x(t)[if x(t) is
aperiodic] or power of signal x(t)[if x(t) is periodic] in the time
domain is equal to the energy or power in the frequency domain.
Then, Parseval’s theorem of Fourier
transform states that
Parseval’s Identity of Fourier Transform
Example:Find the Energy and verify the
Parsvel’s theorem for the Energy Signal
Verify Parsvel’s Theorem?
Correlation
► The correlation of two functions or signals or waveforms is defined as the
measure of similarity between those signals. There are two types of
correlations
• Cross-correlation
• Autocorrelation
❑ The cross-correlation between two different signals or functions or waveforms
is defined as the measure of similarity or coherence between one signal and
the time-delayed version of another signal. The cross-correlation between two
different signals indicates the degree of relatedness between one signal and
the time-delayed version of another signal.
❑ The cross-correlation of energy (or aperiodic) signals and power (or periodic)
signals is defined separately.
Cross-correlation of Energy Signals
Cross-correlation of Energy Signals
► If the energy signals have some similarity. Then, the
cross-correlation between them will have some finite value over the
range τ.
► The variable τ is called the delay parameter or searching parameter or
scanning parameter.
► The time-delay parameter (τ) is the time delay or time shift of one of the two
signals. This delay parameter τ determines the correlation between two
signals.
Cross-correlation of Power Signals
Continue
Auto-Correlation
► The autocorrelation function is defined as the measure of
similarity or coherence between a signal and its time
delayed version. Therefore, the autocorrelation is the
correlation of a signal with itself.
example, autocorrelation of the signal x
[n] = {-1, 2, 1} can be computed
cross-correlation of the digital signals x [n] =
{-3, 2, -1, 1} and y [n] = {-1, 0, -3, 2}
Motivation for the Laplace Transform
Motivation for the Laplace
Transform (continued)
The (Bilateral) Laplace Transform
absolute integrability needed
absolute
integrability
condition
Example #1:
Example #2:
Graphical Visualization of the ROC
Rational Transforms
Many (but by no means all) Laplace transforms of interest to
us are rational functions of s (e.g., Examples #1 and #2; in
general, impulse responses of LTI systems described by
LCCDEs), where
X(s) = N(s)/D(s), N(s),D(s) – polynomials in s
Roots of N(s)= zeros of X(s)
Roots of D(s)= poles of X(s)
Any x(t) consisting of a linear combination of complex
exponentials for t > 0 and for t < 0 (e.g., as in Example #1 and
#2) has a rational Laplace transform.
Example #3
Laplace Transforms and ROCs
Properties of the ROC
More Properties
ROC Properties that Depend on
Which Side You Are On - I
ROC is a right half plane (RHP)
ROC Properties that Depend on
Which Side You Are On -II
ROC is a left half plane (LHP)
ROC Properties that Depend on
Which Side You Are On -II
ROC is a left half plane (LHP)
Example:
Example (continued):
Properties, Properties
If X(s) is rational, then its ROC is bounded by poles or extends to
infinity. In addition, no poles of X(s) are contained in the ROC.
Suppose X(s) is rational, then
a) If x(t) is right-sided, the ROC is to the right of the rightmost pole.
b) If x(t) is left-sided, the ROC is to the left of the leftmost pole.
If ROC of X(s) includes the jω-axis, then FT of x(t) exists.
Example:
Three possible ROCs
Fourier
Transform
exists?
x(t) is right-sided ROC: III No
x(t) is left-sided ROC: I No
x(t) extends for all time ROC: II Yes
Inverse Laplace Transform
Fix σ ∈ ROC and apply the inverse Fourier
But s = σ + jω (σ fixed)⇒ ds =jdω
Inverse Laplace Transforms Via Partial Fraction
Expansion and Properties
Example:
Three possible ROC’s — corresponding to three different signals
Recall
ROC I: — Left-sided signal.
ROC II: — Two-sided signal, has Fourier Transform.
ROC III: — Right-sided
signal.
Properties of Laplace Transforms
Many parallel properties of the CTFT, but for Laplace transforms we need
to determine implications for the ROC
For example:
Linearity
ROC at least the intersection of ROCs of X1(s) and X2(s)
ROC can be bigger (due to pole-zero cancellation)
⇒ ROC entire s-
Time-Domain Differentiation
ROC could be bigger than the ROC of X(s), if there is pole-zero
cancellation. E.g.,
s-Domain Differentiation
Time Shift
