Question
Download Solution PDFConsider Fourier representation of continuous and discrete-time systems. The complex exponentials (i.e., signals), which arise in such representation, have
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFFourier Transform:
Consider the continuous-time signal x(t) and the discrete signal x[n] then Fourier transform is defined as:
For a discrete signal, the DTFT is:
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:
= 1 × 1 + a e-jω + a2 e-2jω + …
Initial values are = ω = 0, -π, π
New values will be ω = 2π, π, 3π
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:
Conclusion:
Complex exponentials and spectrums will have different properties always.
Last updated on Jul 2, 2025
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