Question

Q1 Consider the signal $x(n) = a^n u(n)$, $|a| < 1$. (a) Determine the spectrum $X(\omega)$. (b) The signal $x(n)$ is applied to a decimator that reduces the rate by a factor of 2. Determine the output spectrum. (c) Show that the spectrum in part (b) is simply the Fourier transform of $x(2n)$.

Name: q1 consider the signal xn an un a 1 a determine the spectrum xomega b the signal xn is applied to a decimator that reduces the rate by a factor of 2 determine the output spectrum c show that 11161
Uploaded: 2023-03-14T05:56:58-08:00
Duration: 4 min 6 s
Channel: Madhur L
Description: q1 consider the signal xn an un a 1 a determine the spectrum xomega b the signal xn is applied to a decimator that reduces the rate by a factor of 2 determine the output spectrum c show that 11161

          Q1 Consider the signal $x(n) = a^n u(n)$, $|a| < 1$.
(a) Determine the spectrum $X(\omega)$.
(b) The signal $x(n)$ is applied to a decimator that reduces the rate by a factor of 2. Determine the output spectrum.
(c) Show that the spectrum in part (b) is simply the Fourier transform of $x(2n)$.

q1 consider the signal xn an un a 1 a determine the spectrum xomega b the signal xn is applied to a decimator that reduces the rate by a factor of 2 determine the output spectrum c show that 11161

Added by Alvaro H.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Madhur L

03/14/2023

Step 1

Step 1: (a) The spectrum $X(\omega)$ is the discrete-time Fourier transform (DTFT) of $x(n)$: $$X(\omega) = \sum_{n = -\infty}^{\infty} x(n) e^{-j\omega n} = \sum_{n = 0}^{\infty} a^n e^{-j\omega n}$$ Since this is a geometric series with first term 1 and common Show more…

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Q1 Consider the signal $x(n) = a^n u(n)$, $|a| < 1$. (a) Determine the spectrum $X(\omega)$. (b) The signal $x(n)$ is applied to a decimator that reduces the rate by a factor of 2. Determine the output spectrum. (c) Show that the spectrum in part (b) is simply the Fourier transform of $x(2n)$.