# Chap03 - Chapter 3 Fourier Representations of Signals and...

• Notes
• 140

This preview shows page 1 - 13 out of 140 pages.

Chapter 3 Fourier Representations of Signals and LTI Systems H. F. Francis Lu 2011 Spring: Signal and Systems Ver. 2011.03.27 H. F. Francis Lu 1 / 140
Sec 3.3 Fourier Representation for Four Classes of Signals 2011 Spring: Signal and Systems Ver. 2011.03.27 H. F. Francis Lu 2 / 140
Definition 1 The function x ( t ) defined for all t is said to be periodic if p > 0 such that x ( t + p ) = x ( t ) for all t. The smallest such p is called a fundamental period of x ( t ) . Definition 2 The discrete signal x [ n ] defined for all n Z is said to be periodic if 0 < N Z + such that x [ n ] = x [ n + N ] for all n. The smallest such N is called the fundamental period of x [ n ] . 2011 Spring: Signal and Systems Ver. 2011.03.27 H. F. Francis Lu 3 / 140
Sec 3.5: Continuous-Time Periodic Signals Theorem 1 (Fourier Series) Let x ( t ) be a piecewise continuous function of period T that is defined for all t. The Fourier series (FS) representation of x ( t ) is ˜ x ( t ) = k = -∞ X [ k ] e ı k ω o t where ω 0 = 2 π T and X [ k ] are the Fourier coefficients defined by X [ k ] = 1 T T 0 x ( t ) e - ı k ω o t dt 2011 Spring: Signal and Systems Ver. 2011.03.27 H. F. Francis Lu 4 / 140
Sec. 3.7: Continuous-Time Non-Periodic Signals Theorem 2 (Fourier Transform) Let x ( t ) be a piecewise continuous non-periodic function that is defined for all t. The Fourier Transform (FT) representation of x ( t ) is x ( t ) = 1 2 π -∞ X ( ω ) e ı ω t d ω where X ( ω ) = -∞ x ( t ) e - ı ω t dt 2011 Spring: Signal and Systems Ver. 2011.03.27 H. F. Francis Lu 5 / 140
Sec. 3.6 Discrete-Time Non-Periodic Signals Theorem 3 (Discrete-Time Fourier Transform) Let x [ n ] be a discrete-time non-periodic signal. The Discrete-Time Fourier Transform (DTFT) representation of x [ n ] is x [ n ] = 1 2 π π - π X ( Ω ) e ı Ω n d Ω where X ( Ω ) is defined by X ( Ω ) = n = -∞ x [ n ] e - ı Ω n 2011 Spring: Signal and Systems Ver. 2011.03.27 H. F. Francis Lu 6 / 140
Sec. 3.4 Discrete-Time Periodic Signals Theorem 4 (Discrete Fourier Transform) Let x [ n ] be a discrete-time periodic signal of fundamental period N. The Discrete Fourier Transform (DFT) representation of x [ n ] is x [ n ] = N - 1 n = 0 X [ k ] e ı 2 π nk N where X [ n ] are defined by X [ k ] = 1 N N - 1 n = 0 x [ n ] e - ı 2 π nk N 2011 Spring: Signal and Systems Ver. 2011.03.27 H. F. Francis Lu 7 / 140
Remember This Important Fact DFT is a digital (computer) approximation of FT. Very often you have to be very careful about whether such approximation is accurate. 2011 Spring: Signal and Systems Ver. 2011.03.27 H. F. Francis Lu 8 / 140
2011 Spring: Signal and Systems Ver. 2011.03.27 H. F. Francis Lu 9 / 140
Sec 3.5: Continuous-Time Periodic Signals 2011 Spring: Signal and Systems Ver. 2011.03.27 H. F. Francis Lu 10 / 140
Fourier Series Let x ( t ) be a piecewise continuous function of period T that is defined for all t . The Fourier series representation of x ( t ) is x ( t ) = k = -∞ X [ k ] e ı k ω o t where ω 0 = 2 π T and X [ k ] are the Fourier coefficients defined by X [ k ] = 1 T T 0 x ( t ) e - ı k ω o t dt 2011 Spring: Signal and Systems Ver. 2011.03.27 H. F. Francis Lu 11 / 140
Derivations of Fourier Series Definition 3 Let f ( t ) and g ( t ) be complex-valued functions defined on [ 0 , T ) ; then the following is a well-defined inner product over C .