8 351 Fourier analysis 3

# 8 351 Fourier analysis 3 - Review of Trigonometry 1 Review...

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1 1 Review of Trigonometry 2 Review: Expression with Sine Terms Only Graphically, add these two together. -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Time (sec) Amplitude Acos(wt) Bsin(wt) ( ) 10cos 4 10sin 4 y tt t π = +

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2 3 Review: Expression with Sine Terms Only Find an equivalent expression containing a sine term only. ( ) 10cos 4 10sin 4 y tt t π =+ () * yC s i n t ω φ ( ) 22 * Acos t Bsin t A B sin t ωω += + + 2 2 10 10 14.1 CA B = + = We find: *1 10 tan 45 0.79 rad 10 == = D ( ) 14sin 4 45 yt ° So B 1 * A tan B ϕ = Note: 10*sqrt(2) Note: 45 o 4 Review: Expression with Sine Terms Only -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Time (sec) Amplitude Acos(wt) Bsin(wt)
3 5 Review: Expression with Sine Terms Only -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Time (sec) Amplitude Acos(wt) Bsin(wt) sum 6 Review: Expression with Sine Terms Only -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Time (sec) Csin(wt+f) sum

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4 7 Review: Expression with Sine Terms Only Find an equivalent expression containing a sine term only. () 15cos2 4s in2 y tt t π =+ () * yC s i n t ω φ ( ) 22 * Acos t Bsin t A B sin t ωω += + + 15 4 15.5 CA B = + = We find: *1 15 tan 75 1.3 rad 4 == ° = ( ) 15.5sin 2 1.3 yt So B 1 * A tan B ϕ = Note: larger term dominates Note: maximum is 90 o minimum is 0 o 8 Review: Expression with Sine Terms Only -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Time (sec) Amplitude Acos(wt) Bsin(wt)
5 9 Review: Expression with Sine Terms Only -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Time (sec) Amplitude Acos(wt) Bsin(wt) sum 10 Review: Expression with Sine Terms Only -20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Time (sec) Csin(wt+f) sum

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6 11 Fourier Transforms: Concept 12 Fourier Analysis 0 1 nn n n y( t ) A C sin( t ) ω φ = =+ + Express a time -varying signal in terms of its frequency content. Any signal can be thought of as made up of an infinite sum of sines & cosines of differing periods and amplitudes ( = Fourier series).
7 13 Fourier Analysis Figure from Figliola & Beasley Example 2: RGB pixels on displays Example 1: prism 14 Fourier Transforms: Filtering Application

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8 15 Fourier Analysis Example 3: Lab 2 data 16 Fourier Analysis Example 3: Lab 2 data What is the frequency content of the signal ?
9 17 Fourier Analysis Example 3: Lab 2 data What is the frequency content of the data (information, signal)? The voltage rises in a time of ~1 second, with the rise faster in the first ~1/3 of a second. The signal therefore has significant frequency components up to about 3 Hz. 18 Fourier Analysis Example 3: Lab 2 data What is the frequency content of the noise ? The random fluctuations occur at a wide range of time scales, some almost 1 second long, others much faster. The noise has significant frequency components up to at least tens of Hz. We would need to zoom in on the time scale to see the even faster ones (higher frequencies).

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10 19 Fourier Analysis Example 3: Lab 2 data This is the Fourier transform of a snapshot (~1 sec worth of data in this example) of the time signal.
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## This note was uploaded on 11/28/2010 for the course ENES enes100 taught by Professor Staff during the Spring '10 term at Maryland.

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8 351 Fourier analysis 3 - Review of Trigonometry 1 Review...

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