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Spectral Analysis

AOE 305423 March 2011

Lowe

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Announcements

• Lectures on both Monday, March 28th, and Wednesday, March 30th.– Fracture Testing– Aerodynamic Testing

• Prepare for the Spectral Analysis sessions for next week: http://www.aoe.vt.edu/~aborgolt/aoe3054/manual/inst4/index.html

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What is spectral analysis

• Seeks to answer the question: “What frequencies are present in a signal?”

• Gives quantitative information to answer this question:– The “power (or energy) spectral density”

• Power/energy: Amplitude squared ~V2

• Spectral: Refers to frequency (e.g. wave spectra)• Spectral density: Population per unit frequency ~1/Hz• Units of PSD: V2/Hz

– The phase of each frequency component• How much of the power is sine versus cosine

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Spectral analysis/time analysis

• Given spectral analysis (power spectral density + phase), then we can reconstruct the signal at any and all frequencies:

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Mathematics: Fourier Transforms

• The Fourier transform is a linear transform– Projects the signal onto the orthogonal functions,

sine and cosine:• Two functions are orthogonal if their inner product is

zero:

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Fourier Transform

• We have chosen the functions of interest, now we design the transform:– The Fourier transform works by correlating the

signals of interest to sines and cosines.– Since there are two orthogonal functions that will

fully describe the periodic signal (why?), then a succinct representation is complex algebra. Note:

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Complex Trigonometry

𝐾𝑒𝑟𝑛𝑒𝑙=𝑐𝑜𝑠 (2𝜋 𝑓 𝜏 )−𝑖𝑠𝑖𝑛 (2𝜋 𝑓 𝜏 )=𝑒𝑥𝑝 (− 𝑖2𝜋 𝑓 𝜏 )

𝑐𝑜𝑠 (2𝜋 𝑓 𝜏 )=12 [𝑒𝑥𝑝 (𝑖2𝜋 𝑓 𝜏 )+𝑒𝑥𝑝 (− 𝑖2𝜋 𝑓 𝜏 ) ]

𝑠𝑖𝑛 (2𝜋 𝑓 𝜏 )= 12𝑖 [𝑒𝑥𝑝 (𝑖2𝜋 𝑓 𝜏 )−𝑒𝑥𝑝 (−𝑖2𝜋 𝑓 𝜏 ) ]

Note that time and frequency are called conjugate variables: one is the inverse of the other.

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Fourier transformGenerally, the second moment, or ‘correlation’, of two periodic variables may be written as:

𝑢𝑣= 1𝑇 ∫

0

𝑇

𝑢 (𝑡 )𝑣 (𝑡 )𝑑𝑡

Does this look familiar?

𝑢𝑣=⟨𝑢 ,𝑣 ⟩

A correlation among periodic signals is the inner product of those signals!The Fourier transform is a correlation of a signal with all sines and cosines:

𝐹 [ 𝑠 (𝑡)](𝜔)=∫−∞

∞

𝑠 (𝑡 )𝑒𝑥𝑝 [−𝑖 𝜔𝑡 ]𝑑𝑡

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Fourier transform𝐹 [ 𝑠 (𝑡)](𝜔)=∫

−∞

∞

𝑠 (𝑡 )𝑒𝑥𝑝 [−𝑖 𝜔𝑡 ]𝑑𝑡

• We immediately note:• It yields an answer that is only a function of frequency• It is very closely related to the inner product of the sine/cosine

set of • If the indefinite integration of the kernel is a function time, the

realizability of answers requires a non-infinite limit at (• E

𝐹 [ cos (𝑡 ) ](𝜔)=12 [𝛿 (𝜔−1 )+𝛿 (𝜔+1 ) ]

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Conclusions from cos(t)

• Remember, the Dirac delta function is non-zero only when its input is zero.

• So, is– Zero for all not equal to the angular frequency of

cos(t)– 1 for =1– What is the amplitude of cos(t)?

𝐹 [ cos (𝑡 ) ](𝜔)=12 [𝛿 (𝜔−1 )+𝛿 (𝜔+1 ) ]

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Properties of the Fourier transform

• It is linear: the sum of the FT is the FT of the sum.• The convolution of two signals in time is the product

of those signals in the Fourier domain– Likewise, Fourier domain convolution is equivalent to

time-domain multiplication• The Fourier transform of the derivative of a signal

may be determined by multiplying the transform by • The Fourier transform of the derivative of a signal

may be determined by dividing the transform by

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Digital signals

• Of course, we rarely are so lucky as to have an analytic function for our signal

• More often, we sample, a signal• We can write the Fourier transform in a

discrete manner (i.e., carry out the integration at discrete times/frequencies).

• The Discrete Fourier Transform is

𝐹 [ 𝑠𝑛 ](𝑘)=∑𝑛=0

𝑁−1

𝑠𝑛𝑒𝑥𝑝 [− 𝑖2𝜋𝑁 𝑘𝑛]

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Example: cos(2)

𝐹 [𝑠𝑛 ](𝑘)=∑𝑛=0

𝑁−1

𝑠𝑛𝑒𝑥𝑝 [− 𝑖2𝜋𝑁 𝑘𝑛]

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Multiply:

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Raw Discrete Fourier Transform Results

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FFT and PSD

• The Fast Fourier Transform is an algorithm used to compute the Discrete Fourier Transform based upon

• Beware of scaling:– There are many scalings out there for discrete

Fourier Transforms– There is one easy way to solve this, though,

compute the power spectral density and signal phase.

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PSD Definitions and Signal Phase

• Double-sided spectrum:

• Single-sided spectrum:

• Phase:

𝑢2=∫−∞

∞

𝜙𝑢 (𝜔 )𝑑𝜔

𝑢2=∫0

∞

Φ𝑢 (𝜔 )𝑑𝜔=∫0

∞

2𝜙𝑢(𝜔 ) 𝑑𝜔

𝜑 (𝜔 )=− 𝑡𝑎𝑛−1 ⟨ 𝐼𝑚 {𝐹 [𝑢](𝜔)} /𝑅𝑒𝑎𝑙 {𝐹 [𝑢 ](𝜔)}⟩