Problem 3 An FIR filter is described by the difference equation y/n] = x[n] + a[n-10] %3D (a) Derive its frequency response. (b) Determine its response to the input r[n] = 10 + cos(n)+3 sin (n+). Hints: Use properties of LTI systems; 1+e® = ei% (e=j% + e%) = 2 cos()e; sin a = cos(a-); 10 = 10 e0.". %3D
Problem 3 An FIR filter is described by the difference equation y/n] = x[n] + a[n-10] %3D (a) Derive its frequency response. (b) Determine its response to the input r[n] = 10 + cos(n)+3 sin (n+). Hints: Use properties of LTI systems; 1+e® = ei% (e=j% + e%) = 2 cos()e; sin a = cos(a-); 10 = 10 e0.". %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 3**
An FIR filter is described by the difference equation:
\[ y[n] = x[n] + x[n-10] \]
(a) Derive its frequency response.
(b) Determine its response to the input \( x[n] = 10 + \cos\left(\frac{\pi}{10}n\right) + 3\sin\left(\frac{\pi}{3}n + \frac{\pi}{10}\right) \).
*Hints: Use properties of LTI systems:*
\[ 1 + e^{j\theta} = e^{j\frac{\theta}{2}}(e^{-j\frac{\theta}{2}} + e^{j\frac{\theta}{2}}) = 2\cos\left(\frac{\theta}{2}\right)e^{j\frac{\theta}{2}}; \, \sin(a-\frac{\pi}{2}) = \cos a; \, 10 = 10e^{j0\cdot n}. \]
---
**Problem 4**
Consider an LTI system whose frequency response \( H(\omega) \) is of the following type for some parameters \( a \) and \( b \), and that yields the following input-output pair.
Diagram of \( H(\omega) \):
- The graph is a piecewise linear function with endpoints at \( (-\pi, 0) \) and \( (\pi, 0) \) with a peak value of \( b \) at \( \omega = a \).
Graph of \( x[n] \) showing discrete impulses at:
- \( n = -4, -3, -2, -1, 0, 1, 2, 3, 4 \).
Output \( y[n] \) showing continuous impulses along the \( n \)-axis.
Show that there is enough information to obtain the values of \( a \) and \( b \), and give these values.
*Hint: Express \( x[n] \) in the form \( \alpha_0 e^{j\omega_0 n} + \alpha_1 e^{j\omega_1 n} \) with \( \omega_0 = 0 \) and \( \omega_1 = \pi \).*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e3992ca-2280-40ba-b65a-68dc98c03d5d%2F34e715d1-e5b9-4631-9533-2dab5c7f593f%2F7f2qoo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 3**
An FIR filter is described by the difference equation:
\[ y[n] = x[n] + x[n-10] \]
(a) Derive its frequency response.
(b) Determine its response to the input \( x[n] = 10 + \cos\left(\frac{\pi}{10}n\right) + 3\sin\left(\frac{\pi}{3}n + \frac{\pi}{10}\right) \).
*Hints: Use properties of LTI systems:*
\[ 1 + e^{j\theta} = e^{j\frac{\theta}{2}}(e^{-j\frac{\theta}{2}} + e^{j\frac{\theta}{2}}) = 2\cos\left(\frac{\theta}{2}\right)e^{j\frac{\theta}{2}}; \, \sin(a-\frac{\pi}{2}) = \cos a; \, 10 = 10e^{j0\cdot n}. \]
---
**Problem 4**
Consider an LTI system whose frequency response \( H(\omega) \) is of the following type for some parameters \( a \) and \( b \), and that yields the following input-output pair.
Diagram of \( H(\omega) \):
- The graph is a piecewise linear function with endpoints at \( (-\pi, 0) \) and \( (\pi, 0) \) with a peak value of \( b \) at \( \omega = a \).
Graph of \( x[n] \) showing discrete impulses at:
- \( n = -4, -3, -2, -1, 0, 1, 2, 3, 4 \).
Output \( y[n] \) showing continuous impulses along the \( n \)-axis.
Show that there is enough information to obtain the values of \( a \) and \( b \), and give these values.
*Hint: Express \( x[n] \) in the form \( \alpha_0 e^{j\omega_0 n} + \alpha_1 e^{j\omega_1 n} \) with \( \omega_0 = 0 \) and \( \omega_1 = \pi \).*
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