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# 102_1_lecture16_student - UCLA Fall 2011 Systems and...

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UCLA Fall 2011 Systems and Signals Lecture 16: Time and Frequency Characterization of Systems and Signals November 23, 2011 EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 1

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Time & Frequency Characterization of Systems Why use frequency domain? Alternative signal representation Reveals details that might not be obvious from time-domain inspection Simpliﬁed computation of output Multiplication of transfer functions instead of convolution integral/sum Simple solution to diﬀerential equations Solution is reduced to algebraic procedure Frequency selective ﬁltering can be readily visualized Typically, system speciﬁcations exist in both time & frequency: both domains need to be considered. Note: Throughout the discussion, we perform analysis on continuous signals. However, the same principles hold for discrete-time systems and signals. EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 2
Magnitude / Phase Representation Frequency domain representation is generally complex valued: X ( ) can be expressed in Cartesian form: X ( ) = Real { X ( ) } + j Imag { X ( ) } or in polar (Magnitude-phase) form: X ( ) = | X ( ) | e j ( X ( )) Typically, magnitude-phase form reveals more information. Magnitude - | X ( ) | - describes basic frequency content of the signal: information about amplitudes of complex exponentials that make up the signal. EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 3

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Phase (angle) - X ( ) - describes relative phase of complex sinusoids. Has a substantial eﬀect on nature of the signal! Consider a superposition of three sinusoids: x ( t ) = cos( πt + φ 0 ) + cos(1 . 5 + φ 1 ) + cos(2 + φ 2 ) Relative phases ( φ 0 1 2 ) describe how these sinusoids are combined together. We illustrate this principle with a ﬁgure: left: φ 0 = φ 1 = φ 2 = 0 , middle: φ 0 = 0 , φ 1 = π/ 2 , φ 2 = π , right: φ 0 = 0 , φ 1 = π/ 3 , φ 2 = 2 π/ 3 -5 0 5 -3 -2 -1 0 1 2 3 φ 0 =0 φ 1 =0 φ 2 =0 -5 0 5 -3 -2 -1 0 1 2 3 φ 0 =0 φ 1 = π /2 φ 2 = π -5 0 5 -3 -2 -1 0 1 2 3 φ 0 =0 φ 1 = π /3 φ 2 =2 π /3 EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 4
Although the magnitude of X ( ) in the above example remains the same, time-domain representation changes signiﬁcantly! To further illustrate the importance of phase, consider an extreme example: If x ( t ) is real valued, | X ( ) | = | X ( - ) | . If the time-domain signal is time-reversed, magnitude of X ( ) remains the same - only the phase changes. Yet, the signal in time is clearly diﬀerent. EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 5

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LTI systems change complex amplitudes of frequency components of signal: Y ( ) = H ( ) X ( ) Magnitude of the result is the product of individual magnitudes: | Y ( ) | = | H ( ) || X ( ) | LTI system ”scales” the magnitude of input frequency response Phase of the result is the sum of individual phases: Y ( ) = H ( ) + X ( ) LTI system contributes an additive phase term: H ( ) Even if gain is constant (and even if | H ( ) | = 1 ), phase H ( ) can signiﬁcantly change signal characteristics.
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102_1_lecture16_student - UCLA Fall 2011 Systems and...

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