Consider Fourier representation of continuous and discrete-time systems. The complex exponentials (i.e., signals), which arise in such representation, have

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  1. the same properties always
  2. different properties always
  3. non-specific properties
  4. mostly the same properties

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Option 2 : different properties always
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Fourier Transform:

Consider the continuous-time signal x(t) and the discrete signal x[n] then Fourier transform is defined as:

\(X\left( \omega \right) = \mathop \smallint \nolimits_{ - \infty }^\infty x\left( t \right){e^{ - j\omega t}}dt\)

For a discrete signal, the DTFT is:

\(\dot X\left( {{e^{j\omega }}} \right) = \mathop \sum \limits_{n = - \infty }^\infty x\left[ n \right]{e^{ - j\omega n}}\)

It is periodic from ω = -π to π with the total period as 2π

It gives the same values for ω, ω + 2π, ω + 4π …. etc.

Whereas CTFT is different for different values of ‘ω’

CTFT is also non-periodic.

Consider a discrete time signal x[n] = an u[n] |a| < 1

DTFT of this signal is calculated as:

\(X\left( {{e^{j\omega }}} \right) = \mathop \sum \limits_{n = - \infty }^\infty {a^n}u\left[ n \right]{e^{ - j\omega n}}\)

\( = \mathop \sum \limits_{n = 0}^\infty {a^n}{e^{ - j\omega n}}\)

= 1 × 1 + a e-jω + a2 e-2jω + …

\(X\left( {{e^{j\omega }}} \right) = \frac{1}{{1 - a{e^{ - j\omega }}}}\)

\(X\left( {{e^{j\left( { - \pi } \right)}}} \right) = \frac{1}{{1 - a{e^{ - j\left( { - \pi } \right)}}}} = \frac{1}{{1 - a{e^{j\pi }}}} = \frac{1}{{1 - a\left( { - 1} \right)}} = \frac{1}{{1 + a}}\)

\(X\left( {{e^{j0}}} \right) = \frac{1}{{1 - a{e^{j.0}}}} = \frac{1}{{1 - a}}\)

\(X\left( {{e^{j\pi }}} \right) = \frac{1}{{1 - a{e^{j\pi }}}} = \frac{1}{{1 - a\left( { - 1} \right)}} = \frac{1}{{1 + a}}\)

Initial values are = ω = 0, -π, π

New values will be ω = 2π, π, 3π

\(X\left( {{e^{j\pi }}} \right) = \frac{1}{{1 - a{e^{j\pi }}}} = \frac{1}{{1 - a\left( { - 1} \right)}} = \frac{1}{{1 + a}}\)

\(X\left( {{e^{j2\pi }}} \right) = \frac{1}{{1 - a{e^{j2\pi }}}} = \frac{1}{{1 - a\left( 1 \right)}} = \frac{1}{{1 - a}}\)

\(X\left( {{e^{j3\pi }}} \right) = \frac{1}{{1 - a{e^{j3\pi }}}} = \frac{1}{{1 - a\left( { - 1} \right)}} = \frac{1}{{1 + a}}\)

So, the same values are repeating for the adding period.

Consider a continuous signal x(t) = e-at u(t).

CTFT of this signal is calculated as:

\(X\left( \omega \right) = \mathop \smallint \nolimits_0^\infty {e^{ - at}}{e^{ - j\omega t}}dt\)

\( = \frac{1}{{a + j\omega }}\)

\(\left| {X\left( \omega \right)} \right| = \frac{1}{{\sqrt {{a^2} + {\omega ^2}} }}\) and spectrum is:

F5 Shubham 23-10-2020 Swati D5

Conclusion:

Complex exponentials and spectrums will have different properties always.
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