Explain First Shift Theorem &amp; its properties?

Name: Explain First Shift Theorem &amp;amp; its properties?
Uploaded: 2022-02-23T17:22:57+00:00
Description: Explain First Shift Theorem &amp;amp; its properties?

Question

Explain First Shift Theorem &amp;amp; its properties?

nitruraviX · Accepted Answer

Step 1  First Shift Theorem:  If L{f(t)}=F(s), then L{eatf(t)}=F(s−a)  The proof of the First shift theorem follows from the definition of Laplace transform. It is known that,  L{f(t)}=∫0∞e−stf(t)dt  Step 2  Then,  L{eatf(t)}=∫0∞eat⋅e−stf(t)dt  =∫0∞e(a−s)tf(t)dt  =∫0∞e−(s−a)tf(t)dt  =F(s−a)  Thus, if the Laplace transform of function f(t) is known, then we can find the Laplace transform of eatf(t) using First shift theorem.  Step 3  For example,  It is known that, L{sin⁡3t}=3(s2+9)  Then,  L{e2t⋅sin⁡3t}=3(s−2)2+9 by first shift theorem.

RizerMix · Accepted Answer

Let f(t) be a function with Fourier Transform F(ω). If f(t) is shifted by a time t0, resulting in g(t)=f(t−t0), then the Fourier Transform of g(t), denoted as G(ω), is given by:G(ω)=e−jωt0·F(ω)In this equation, j represents the imaginary unit, ω denotes the angular frequency, t0 is the amount of shift in the time domain, and e is the base of the natural logarithm.Properties of the First Shift Theorem:1. Timeshift: The First Shift Theorem states that shifting a function in the time domain corresponds to multiplying its Fourier Transform by a complex exponential term with a frequency-dependent phase shift.2. Frequency&amp;nbsp;shift: Conversely, the theorem also implies that a shift in the frequency domain corresponds to multiplication by a complex exponential term with a time-dependent phase shift in the time domain.3. Symmetry: The First Shift Theorem exhibits symmetry between the time and frequency domains. Shifting a function in one domain causes a corresponding shift in the other domain.4. Linearity: The First Shift Theorem holds for linear combinations of functions. If f(t) and h(t) have Fourier Transforms F(ω) and H(ω) respectively, then the Fourier Transform of af(t)+bh(t), where a and b are constants, is given by aF(ω)+bH(ω).The First Shift Theorem is a crucial property of the Fourier Transform and finds applications in various fields such as signal processing, communications, image processing, and quantum mechanics. It enables the analysis and manipulation of signals in both the time and frequency domains, providing valuable insights into the behavior of systems and signals.

Vasquez · Accepted Answer

The First Shift Theorem is a property of the Fourier Transform, which relates the effect of shifting a function in the time domain to its Fourier Transform in the frequency domain. Mathematically, it can be stated as follows:

First Shift Theorem:

Let

f (t)

be a function with Fourier Transform

F (ω)

. If

f (t)

is shifted by a time

t_{0}

to form

f (t - t_{0})

, then the Fourier Transform of the shifted function is given by:

ℱ [f (t - t_{0})] = e^{- j ω t_{0}} F (ω)

where

ℱ

represents the Fourier Transform operator and

j

is the imaginary unit.
The First Shift Theorem has the following properties:

1. Time-domain shift property:

If

f (t)

is shifted by

t_{0}

to form

f (t - t_{0})

, then the function in the frequency domain is multiplied by a complex exponential

e^{- j ω t_{0}}

.

2. Frequency-domain shift property:

If

F (ω)

is the Fourier Transform of

f (t)

, then the Fourier Transform of

f (t - t_{0})

is obtained by shifting

F (ω)

by

t_{0}

in the time domain. This can be expressed as:

ℱ [f (t - t_{0})] = F (ω) e^{- j ω t_{0}}

3. Phase shift property:

The First Shift Theorem also introduces a phase shift in the frequency domain due to the time-domain shift. The phase shift is given by

e^{- j ω t_{0}}

.

4. Linearity property:

The First Shift Theorem holds true for linear combinations of shifted functions. If

f (t) = \sum_{i} a_{i} g_{i} (t)

, where

g_{i} (t)

are shifted functions, then the Fourier Transform of

f (t)

can be obtained by shifting the Fourier Transform of

g_{i} (t)

and taking the linear combination.
These properties make the First Shift Theorem a powerful tool for analyzing the effects of time-domain shifts on the frequency content of a signal.

Don Sumner · Accepted Answer

The theorem states that if x(t) is a function with Fourier transform X(ω), and we shift x(t) by a time delay t0, resulting in x(t−t0), then the Fourier transform of the shifted function is given by:ℱ{x(t−t0)}=X(ω)·e−jωt0In this equation, ℱ denotes the Fourier transform operator, and j represents the imaginary unit. The expression e−jωt0 represents a complex exponential term that introduces a phase shift proportional to the time delay t0.The properties of the First Shift Theorem include:1. Time delay in the time domain corresponds to phase shift in the frequency domain.2. The magnitude spectrum of the Fourier transform remains unchanged; only the phase spectrum is affected.3. The frequency content of the signal remains the same, but it appears at a different time instant.4. Shifting the signal in the time domain does not affect its bandwidth or energy distribution in the frequency domain.5. The inverse Fourier transform of the shifted spectrum will yield the original function shifted by t0.The First Shift Theorem is a valuable tool in signal processing and communication systems, allowing us to analyze the effects of time shifts on the frequency content of a signal.

Explain First Shift Theorem & its properties?

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