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lec18_handout

# lec18_handout - From last time… • Op-amp circuits –...

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Unformatted text preview: From last time… • Op-amp circuits – “Active filters” – Can cause less loading than equivalent passive filter IF RF Ei Ri Ii E+ Ei Eo E- EEd E+ RF Ri RF ZF CF Ei Ri Ii Ri Ei E Ed E+ RF Zi Ci Ii E Ed E+ Eo ME365 Spectrum Analysis Eo 1 Spectrum Analysis • Signals • Periodic Signals – Fourier Series Representation of Periodic Signals • Frequency Spectra – Amplitude and Phase Spectra of Signals – Signal Through Systems - a Frequency Spectrum Perspective • Non-periodic Signals - Fourier Transform • Random Signals - Power Spectral Density sin cos sin cos cos cos cos ∓ sin sin sin sin cos ME365 Spectrum Analysis ∓ sin cos sin cos 2 2 cos ∓sin 2 Signals Input Signal x(t) Linear System G(jω) Output Signal y(t) Can be characterized by its Frequency Response Function G(jω) = | G(jω)|ejArg[G(jω) ] • Signals can be categorized as: – Periodic Signals – Non-Periodic Signals (well defined) – Random Signals ⇒ Would like to characterize signals in the frequency domain! ME365 Spectrum Analysis 3 Periodic Signals - Fourier Series Any periodic function x(t), of period T, can be represented by an infinite series of sine and cosine functions of integer multiples of its fundamental frequency ω1 = 2π/T . x (t ) = A0 ∞ + ∑ Ak cos ( kω 1 t ) + Bk sin ( kω 1t ) 2 k =1 [ where x (t ) = x (t + T ) and ω 1 = 2π [rad /sec ] and T ⎧ A0 : Amplitude of the DC component ⎪2 ⎨ 2π ⎪kω 1 = k T : the kth harmonic frequency ⎩ ⎧ A = 2 T x (t )dt ⎪ 0 T ∫0 T ⎪ 2 Fourier Coefficients: ⎨ Ak = T ∫ x (t ) cos( kω 1t )dt 0 ⎪ T 2 ⎪Bk = T ∫0 x (t )sin( kω 1t )dt ⎩ ME365 Spectrum Analysis 4 Fourier Series Ex: triangle signal with period T sec. 8 2 Take the first six terms and let A = 10, T = 0.1: x ( t ) = 8 .105 cos( 20 π t ) + 0 . 901 cos( 3 × 20 π t ) + 0 . 324 cos( 5 × 20 π t ) + 0 .165 cos( 7 × 20 π t ) cos 1 + 0 .1 cos( 9 × 20 π t ) + 0 . 067 cos( 11 × 20 π t ) Summation of Input Signals 10 8 T Acceleration A 6 4 2 0 -2 0 -4 -6 2T 0.05 0.1 -8 -10 0.15 0.2 Time (sec) ME365 Spectrum Analysis 5 Fourier Series Pointers for Calculating the Fourier Coefficients: • A0 /2 represents the average of the signal x(t). It contains the “DC” (zero frequency) component of the signal. 1 3 1 2 t 2 2 t 4 -1 2 • When calculating the Fourier coefficients, changing the integral limits will not affect the results, as long as the integration covers one period of the signal. x(t) T 2 T/2 3T/2 T -T/2 t 2 cos cos 2 2 T ME365 Spectrum Analysis 6 Fourier Series Pointers for Calculating the Fourier Coefficients: • Odd functions (signals), for which x(−t) = − x(t), will only contain sine x(t) terms, i.e. Ak = 0, for k = 0, 1, 2, ... . if sin sin 0 sin ⇒ t T T • Even functions (signals), for which x(−t) = x(t), will only contain cosine terms, i.e. Bk = 0, for k = 1, 2, ... . if cos A cos ⇒ 0 cos 2 T 2T ME365 Spectrum Analysis 7 Fourier Series Ex: Half rectified sine wave sin x (t) 0 E −π/ω 0 π/ω 2π/ω Symbolic: Q: What is the period, T, of this signal? What is the Fundamental Frequency of this signal? 0 time (t) Numeric: Let ω = 1 [rad/s], E = 1 [V]. Q: Expand the signal into its Fourier series. → Find the corresponding Fourier Coefficients! ME365 Spectrum Analysis 8 Fourier Series Numeric: Symbolic: Calculate Fourier Coefficients: 2 2 2 2 cos sin 2 2 sin cos ME365 Spectrum Analysis 9 Fourier Series Numeric: Symbolic: Calculate Fourier Coefficients: 2 sin ME365 Spectrum Analysis 2 2 sin sin 10 Fourier Series • Fourier Series Representation: ME365 Spectrum Analysis 11 Fourier Series Ex: E −2 x (t) 4 2 4 2 2 time (t) Q: What is the period, T, of this signal? What is the Fundamental Frequency of this signal? Q: Expand the signal into its Fourier series. ME365 Spectrum Analysis 12 Fourier Series 2 2 cos ME365 Spectrum Analysis 13 Fourier Series • Different Fourier Series Representations: 2 cos sin 2 2 ME365 Spectrum Analysis 14 Fourier Series • Fourier Series in Complex Form: cos 2 Recall cos sin sin where 1 ME365 Spectrum Analysis 15 Fourier Series 2 2 2 cos sin tan cos tan sin 1 2 1 2 cos sin ME365 Spectrum Analysis ∠ 1 2 tan 2 ⋅ Re 2 ⋅ Im 16 Fourier Series Ex: Write the following periodic signal in a sine and cosine series form and plot its magnitude and phase vs frequency plot. Ex: Write the following periodic signal in a complex Fourier series form and plot its magnitude and phase vs frequency plot. 4 sin 24 5 sin 72 3 cos 24 12cos 72 ME365 Spectrum Analysis 17 Frequency Spectra • Another interpretation of the Fourier Series representation of periodic signals is that the combination of the Fourier coefficients and their corresponding harmonics characterizes the periodic signal. A periodic signal can be represented by: 2 cos – Time Domain: Signal Amplitude vs Time x (t) – Frequency Domain: Fourier coefficient Amplitude and Phase vs Frequency. Time (t) Phase Spectrum Amplitude Spectrum θk Mk 0 ω1 ME365 Spectrum Analysis 2ω1 3ω1 (ω) 4ω1 Frequency 0 ω1 2ω1 3ω1 (ω) 4ω1 Frequency 18 Frequency Spectra Ex: Plot the Amplitude and Phase Spectra of the following signal: 5 4 cos 40 Ex: What is the fundamental frequency of the signal in the previous example? 3 3 cos 60 2 6 sin 100 Amplitude Spectrum 4 Mk Ex: What is the output signal of a low pass 0 40π 60π 100π Frequency(ω) Phase Spectrum θk 0 40π 60π filter if x(t) in the previous example is passed through a passive low-pass filter with gain 1 and time constant 0.1 sec? 100π Frequency(ω) ME365 Spectrum Analysis 19 Signals Through Systems Linear System G(jω) Input Periodic Signal x(t) 2 Output Signal y(t) ? cos Can be characterized by its Frequency Response Function G(jω) = | G(jω)|ejArg[G(jω) ] 2 ME365 Spectrum Analysis cos 20 Signals Through Systems Ex: An inkjet nozzle "firing" signal has the following amplitude (Mk) and phase spectra (ψk). (This is the spectrum generated from the sine with phase shift form of the Fourier Series.) Write down the Fourier Series of this signal, xA(t). ME365 Spectrum Analysis 21 Signals Through Systems Ex: After going through the PC board on the pen carriage, the spectrum plots of the signal looked like: Write down the augmented Fourier Series of the new signal xB(t). ME365 Spectrum Analysis 22 Signals Through Systems Ex: If the signal xB(t), is to pass through a filter with the following frequency response function: 1 0.00318 1 Write down the Fourier Series of the output signal, xC(t). Plot the amplitude and phase spectrum of the output signal, xC(t). ME365 Spectrum Analysis 23 ...
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