8 351 Fourier analysis 3

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

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon
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 = +
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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)
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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)
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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).
Background image of page 6
7 13 Fourier Analysis Figure from Figliola & Beasley Example 2: RGB pixels on displays Example 1: prism 14 Fourier Transforms: Filtering Application
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 ?
Background image of page 8
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).
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 10
Image of page 11
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 26

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

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online