Lesson 15 - Convolution Theorem

# Lesson 15 - Convolution Theorem - When convolution goes...

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Lesson 15 Convolution Theorem Challenge 14 Lesson 15/Lecture 17 – Convolution Theorem Challenge 15 Monday: Tentative – Project II musings When convolution goes very wrong.

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Lesson 15 Side Bar The core signal processing domains are: communication control audio/acoustics video/imaged radar/sonar/defense geo-science biomedicine instrumentation Military Instrumentation Industrial Consumer Computer Wireline Wireless
Lesson 15 Challenge 14 A 5-sample moving average (MA) filter is presented with a 10-sample unit step and 10-sample zero mean Gaussian random noise process ( signal ), all sampled at a sample rate of 1,000 Sa/s. In each case there is a transient “build up,” “build down,” and what is called the “stead-state” phase. Q: What is the length, in seconds, of each of these phases? Averaging Filter Noise

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Lesson 15 Challenge 14 The input signal is of length N 1 =10 samples (10ms). The filter is of length N 2 =5 samples (5ms). The convolution sum is of length N 1 +N 2 -1 =14 samples (14 ms). ------------------------------------------------------------------------------------- The “transient build up” lasts for N 2 -1= 4 samples ( 4 ms ). The “transient build down” lasts for N 2 -1= 4 samples ( 4 ms ). “Steady state” behavior persists for 14 – 4 - 4 = 6 samples ( 6 ms ). Averaging Filter
Lesson 15 Challenge 14 Transient “build up”  mode Samples 0, 1, 2,  and 3; length 4 Transient “build  down” mode Samples 10, 11,  12, and 13; length  4 “Steady state mode. Samples 4, 5, 6, 7, 8, and 9; length 6 Step-response 0.2 1.0 4 4 6 All 5  multipliers  doing work . Pulse input

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Lesson 15 Challenge 14 Samples 0, 1,  2, and 3;  length 4 Samples 10, 11,  12, and 13;  length 4 Samples 4, 5, 6,  7, 8, and 9;  length 6 Zero-mean  additive noise response All 5  multipliers  doing work . Random input
Lesson 15 Back to Convolution 0.5 1 1.5 2 2.5 3 x 10 4 -60 -50 -40 -30 -20 -10 Requirements Drop 85% within 0.07 seconds At a sample rate of 44.1kSa/s 0.07 seconds translates to ~3100 samples 85% attenuation as -16.5 dBFS (20*log10(0.15) What are you actually measuring? Clap test

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Lesson 15 Lesson 15 Convolution Theorem
Lesson 15 Convolution Motivation Why is normal noise normal? (bold claim: the physical world is a natural lowpass filter?) 1 0 -1 PDF 1 0 -1 h[k]  (channel) Random (uniform)  signal x[k]  (binary valued) y 1 [k]= x[k]*h[k] h[k] x[k] y 1 [k] PDF

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Lesson 15 Back to Convolution h[k]  (channel) y 1 [k]  (previous  output) y 2 [k]= y 1 [k]*h[k] 1 0 -1 PDF 1 0 -1 h[k] y 1 [k] y 2 [k] PDF
Lesson 15 Lesson 15 The convolution sum is defined in terms of the system’s impulse response h [ k ], and input x [ k ]. Suppose

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## This note was uploaded on 01/26/2012 for the course ECON 101 taught by Professor Flah during the Spring '10 term at Punjab Engineering College.

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Lesson 15 - Convolution Theorem - When convolution goes...

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