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16 Pages

### Fourier

Course: CH 2007, Fall 2009
School: Oregon
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Word Count: 1153

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two Given periodic functions y=Asin1t Determine whether 1=2 First, multiply the functions together: 1 yz = Asin 1t sin 2 t = A{cos ( 1t 2 t) cos( 1t + 2 t )} 2 z=sin2t Next, average over the signals for a long time If 1 and 2&gt;0, cos(1t+2t) averages to zero If 1 2, cos(1t- 2t) averages to zero If 1 =2, cos(1t-2t) = 1 1 A if 1 = 2 Average = 2 otherwise 0 Now take the next most complicated case:...

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two Given periodic functions y=Asin1t Determine whether 1=2 First, multiply the functions together: 1 yz = Asin 1t sin 2 t = A{cos ( 1t 2 t) cos( 1t + 2 t )} 2 z=sin2t Next, average over the signals for a long time If 1 and 2>0, cos(1t+2t) averages to zero If 1 2, cos(1t- 2t) averages to zero If 1 =2, cos(1t-2t) = 1 1 A if 1 = 2 Average = 2 otherwise 0 Now take the next most complicated case: suppose y=Asin(1t+) yz = Asin( 1t + ) sin 2 t z=sin2t Again multiply the two signals: = A[sin 1 tcos + cos 1tsin ] sin 2 t = Acos sin 1t sin 2 t + A sin cos 1t sin 2 t = a1 sin 1 tsin 2t + a 2 cos 1t sin 2 t = 1 a1 {cos( 1 2 )t cos(1 + 2 )t} 2 1 + 2 a2 { sin( 1 2 )t sin( 1 + 2 )t} 1 a1 Average = 2 0 if 1 = 2 otherwise The other part of the amplitude of y=Asin(1t+) can be determined by comparing with z=cos2t: yz = Asin( 1t + ) cos 2 t = A[sin 1 tcos + cos 1tsin ] cos 2 t = Acos sin 1t cos 2 t + A sin cos 1 tcos 2t = a1 sin 1 tcos 2 t + a2 cos 1tcos 2t = 1 a1 {sin( 1 2 )t sin( 1 + 2 )t} 2 1 + 2 a2 { cos( 1 2 )t cos( 1 + 2 )t} 1 a2 if 1 = 2 Average = 2 otherwise 0 It takes two functions to determine the amplitude of a periodic function with arbitrary phase. The sine and cosine transforms The procedure above is formally defined by the sine and cosine transforms of a function h(t): C( f ) = S( f ) = h(t) cos(2ft)dt h(t) sin (2 ft )dt The Fourier transform Using complex notation with we can write e 2ift = cos(2ft ) + i sin(2ft ) H ( f ) = h(t )e 2ift 2ift dt H(f) is the Fourier transform of h(t). It turns out that this transform can be inverted simply: h(t ) = H ( f )e 2ift 2ift dt H(f) and h(t) are said to be a Fourier transform pair. Fourier transforms are convenient for representing periodic functions as sines and cosines decomposing periodic functions into harmonic contributions A Fourier transform represents data in a reciprocal domain: time frequency length wavenumber This becomes useful if an experiment is easy to perform in one domain but analysis is more convenient in the other. Examples Ion resonance mass spectrometer Fourier transform infrared spectrometers Fourier transform NMR spectrometers Digital signal processing for communication devices Encoding and compression of audio signals The ion cyclotron spectrometer The resonant frequency of an ion moving in a circular orbit in a magnetic field is for a single kind of ion, the signal will be a periodic function with frequency f = q B 2m B = 2f = q m For several species, the signal will be a sum of several such signals. Fourier A transform will provide information about the number and identity of the several kinds of ions. Forms of the F.T. To extract all the frequency and phase information from a time-dependent signal, use the full Fourier transform: 2ift 2ift H ( f ) = h(t )e dt If you only care about the amplitude and not the phase, you don't need the (complex) H(f). Instead, specify the power density as this also is called the one-sided power spectral density, the power function or the power spectrum. P ( f ) = H ( f ) + H ( f ) 2 2 2 2 Properties of the Fourier Transform Linearity: if h(t ) H ( f) ,k (t) K ( f ) then a h(t ) + b k (t) a H ( f ) + b K( f ) Time shift/frequency shift: if h(t ) H ( f ) then h(t t ) H f e 2ift0 0 () h (t )e 2if0 t H ( f f0 ) Properties of the Fourier Transform Time and Frequency Scaling : if h(t ) H ( f ) 1 f 1 t then h( at) H and h H ( af ) a a a a The Convolution Theorem: if h(t ) H ( f ) g(t) G( f ) then g(t) h(t) G( f ) H ( f ) G ( f ) H( f ) = G( )H ( f )d The discrete transform Real experiments sample at finite rates and for finite duration. For the sampling interval t, H ( fn ) h(t)e n and for N samples, N 1 N 1 k= 0 k= 0 2 ifn t 2 ifn t t H ( fn ) t hkke n n N N fn = ,n= L Nt 2 2 2ikn 2ikn N N tk = kt,k = 0LN 1 The discrete Fourier transform is defined as H ( fn ) n Hn = = hk e n k t k= 0 k= 0 N1 N1 2ikn 2ikn N N and its (one-sided) inverse is 1 hkk = H n e n N n= 0 n= 0 N 1 N 1 2ikn 2ikn N N The Limit of the Fourier Transform For a gi...

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Oregon - CH - 417
Given two periodic functionsy=Asin1t Determine whether 1=2 First, multiply the functions together:1 yz = Asin 1t sin 2 t = A{cos ( 1t 2 t) cos( 1t + 2 t )} 2z=sin2t Next, average over the signals for a long time If 1 and 2&gt;0, cos(1t+2t) ave
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The nature of logic circuitsA logic circuit is an electronic device used to mimic the operations of symbolic logic. Symbolic logic consists of expressions which contain variables and operations, and which may assume values of &quot;true&quot; or&quot;false.&quot; In th
Oregon - CH - 417
The nature of logic circuitsA logic circuit is an electronic device used to mimic the operations of symbolic logic. Symbolic logic consists of expressions which contain variables and operations, and which may assume values of &quot;true&quot; or&quot;false.&quot; In th
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Writing a lab reportThe first thing to realize in preparing a technical paper is that there is a prescribed literary form for such a document. Like a haiku or a sonnet, a scientific paper must adhere to certain standards of organization and presenta
Oregon - CH - 2007
Active and passive circuit elementsThe circuit elements we have encountered up to now have been, for the most part, passive circuit elements. These include, for example, resistors and capacitors. They alter the flow of electrons through a circuit, b
Oregon - CH - 417
Active and passive circuit elementsThe circuit elements we have encountered up to now have been, for the most part, passive circuit elements. These include, for example, resistors and capacitors. They alter the flow of electrons through a circuit, b
Oregon - CH - 2007
Compressibility of a real gasThe compressibility factor of a gas (and, by extension, any fluid) is defined asPVm Z= RTFor an ideal gas, the compressibility factor is always 1. For a real gas, the deviation of the compressibility factor from 1 is
Oregon - CH - 417
Compressibility of a real gasThe compressibility factor of a gas (and, by extension, any fluid) is defined asPVm Z= RTFor an ideal gas, the compressibility factor is always 1. For a real gas, the deviation of the compressibility factor from 1 is
Oregon - CH - 2006
An overview of GC/MSAutomatic liquid sampler (ALS) Mass spectrometerGas chromatographThe autosamplerThis is the only part of the apparatus that actually appears to do anything. Its function is to inject a measured sample onto the GC column.Th
Oregon - CH - 418
An overview of GC/MSAutomatic liquid sampler (ALS) Mass spectrometerGas chromatographThe autosamplerThis is the only part of the apparatus that actually appears to do anything. Its function is to inject a measured sample onto the GC column.Th
Oregon - CH - 2006
Base catalyzed ester hydrolysisWe start with what you learned in organic chemistry:R C O + OHkOR' k+ O R C OR' OH-activated complexk'R C O + OR'very fastR C O + R'OHOHOBase catalyzed ester hydrolysis.Symbolically,k' AB C + D A + B
Oregon - CH - 418
Base catalyzed ester hydrolysisWe start with what you learned in organic chemistry:R C O + OHkOR' k+ O R C OR' OH-activated complexk'R C O + OR'very fastR C O + R'OHOHOBase catalyzed ester hydrolysis.Symbolically,k' AB C + D A + B
Oregon - CH - 2006
Solutions of rate equationsThe analytic method:dN = kN Write the differential equation dt dN = kdt Rearrange N N Integrate ln = kt N0SimplifyN = N0 e ktThe integration step depends, among other things, on having only one variable on each
Oregon - CH - 418
Solutions of rate equationsThe analytic method:dN = kN Write the differential equation dt dN = kdt Rearrange N N Integrate ln = kt N0SimplifyN = N0 e ktThe integration step depends, among other things, on having only one variable on each
Oregon - CH - 2006
Photochemical processesAbsorption of light (creation of an electronically excited state) Relaxation (thermalization of vibrational energy within that state) Emission from the original excited state (fluorescence) Thermalization of electronic energy
Oregon - CH - 418
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Oregon - CH - 2007
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The Boltzmann distributionThe Boltzmann distribution predicts the probability that a molecule will be found in a particular state with a given energy if the molecule is in thermal equilibrium with a constant temperature bath: EiPi ! e&quot;kTThe
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Oregon - CH - 2006
Phosphatase-catalyzed hydrolysisIn this experiment you will monitor the hydrolysis of a phosphate ester. This hydrolysis is catalyzed by a phosphatase enzyme (typically one derived from wheat germ or potatoes). This reaction is extremely important i
Oregon - CH - 418
Phosphatase-catalyzed hydrolysisIn this experiment you will monitor the hydrolysis of a phosphate ester. This hydrolysis is catalyzed by a phosphatase enzyme (typically one derived from wheat germ or potatoes). This reaction is extremely important i
Oregon - CH - 2007
Solutions of rate equationsThe analytic method:dN = ! kN Write the differential equation dt dN = ! kdt Rearrange N N Integrate ln = !kt N0SimplifyN = N0 e! ktThe integration step depends, among other things, on having only one variable on eac
Oregon - CH - 418
Solutions of rate equationsThe analytic method:dN = ! kN Write the differential equation dt dN = ! kdt Rearrange N N Integrate ln = !kt N0SimplifyN = N0 e! ktThe integration step depends, among other things, on having only one variable on eac
Oregon - CH - 2006
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Oregon - CH - 418
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Oregon - CH - 2007
Acid phosphatase catalyzed hydrolysis of p-nitrophenol phosphateMonitor p-nitrophenol vs. time Convert p-nitrophenol concentration to a reaction rate Perform the Lineweaver-Burk t Compare with literatureMonitoring p-nitrophenolIn the p-nitropheno
Oregon - CH - 418
Acid phosphatase catalyzed hydrolysis of p-nitrophenol phosphateMonitor p-nitrophenol vs. time Convert p-nitrophenol concentration to a reaction rate Perform the Lineweaver-Burk t Compare with literatureMonitoring p-nitrophenolIn the p-nitropheno
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EtOAc.nb1The basic integral:In[9]:=Out[9]=1 x Ha - xL Hb - xLLog@-a + xD - Log@-b + xD a-bEvaluate at the limits of integration:In[10]:= Out[10]=Log@-a + xD - Log@-b + xD Log@-a + 0D - Log@-b + 0D - a-b a-b Log@-aD -
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EtOAc.nb1The basic integral:In[9]:=Out[9]=1 x Ha - xL Hb - xLLog@-a + xD - Log@-b + xD a-bEvaluate at the limits of integration:In[10]:= Out[10]=Log@-a + xD - Log@-b + xD Log@-a + 0D - Log@-b + 0D - a-b a-b Log@-aD -
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Modeling Reaction RatesThe procedure for modeling chemical reaction rates generally follows these steps: Postulate a chemical modeln1 R1 + n2 R2 + ProductWrite the related rate equationd[Product] n1 n2 = [R1 ] [R2 ] dtIntegrate the differe
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