延安大学：《数字信号处理》课程PPT教学课件（DigitalSignal Processing，DSP）Chapter 2 The Discreete-Time Fourier Transform（DTFT）

2 Discrete-Time fourier Transform

2 Discrete-Time Fourier Transform

Introduce o The signals and systems can be analyzed in time-domain or frequency-domain o In time-domain, any arbitrary sequence can be represented as a weighted linear combination of delayed unit sample sequence, then the input-output relationship of LTI system can be obtained o The frequency-domain representation of a discrete-time sequence is also discussed in this chapter

Introduce ⚫ The signals and systems can be analyzed in time-domain or frequency-domain. ⚫ In time-domain, any arbitrary sequence can be represented as a weighted linear combination of delayed unit sample sequence, then the input-output relationship of LTI system can be obtained. ⚫ The frequency-domain representation of a discrete-time sequence is also discussed in this chapter

Introduce o In many applications, it is convenient to consider an alternate description of a sequence in terms of complex exponential sequences. This leads to a particularly useful representation of discrete-time sequences and certain discrete-time systems in frequency domain

Introduce ⚫ In many applications, it is convenient to consider an alternate description of a sequence in terms of complex exponential sequences. This leads to a particularly useful representation of discrete-time sequences and certain discrete-time systems in frequency domain

2.1 The Continuous-Time Fourier Transform We begin with a brief review of the continuous-time Fourier transform, a frequency-domain representation of a continuous-time signal, and its properties, as it will provide a better understanding of the frequency-domain representation of the discrete-time signals and systems in addition to pointing out the major differences between these two transform

2.1 The Continuous-Time Fourier Transform We begin with a brief review of the continuous-time Fourier transform, a frequency-domain representation of a continuous-time signal, and its properties, as it will provide a better understanding of the frequency-domain representation of the discrete-time signals and systems, in addition to pointing out the major differences between these two transform

2.1 The continuous-Time fourier transform e Definition of continuous-time ft Continuous-time Fourier transform(CTFT) xa(jQ)=xa( e-c Inverse continuous-time Fourier transform(ICTFT) X(iQe/ds 2丌

2.1 The Continuous-Time Fourier Transform • Definition of continuous-time FT ( ) ( ) j t X j x t e dt a a  −  −  =  ( ) ( ) 1 2 j t a a x t X j e d    − =    Continuous-time Fourier transform (CTFT) Inverse continuous-time Fourier transform (ICTFT)

2.1 The Continuous-Time fourier transform Definition of continuous-time ft The ctft can also be expressed in polar form as X2(g2)=|Xx(2)e where magnitude spectrum ea()=arg Xa(j Q2) phase spectrum

2.1 The Continuous-Time Fourier Transform • Definition of continuous-time FT The CTFT can also be expressed in polar form as ( ) ( ) j a ( ) X j X j e a a    =  where a a (t X j ) =  arg ( ) magnitude spectrum phase spectrum X j a ( )

2.1 The Continuous-Time fourier transform Definition of continuous-time ft Dirichlet conditions (a The signal has a finite number of finite discontinuous and a finite number of maxima and minima in any finite interval (b) The signal is absolutely integrable, that is xa (tdt < oo

2.1 The Continuous-Time Fourier Transform • Definition of continuous-time FT Dirichlet conditions: (a) The signal has a finite number of finite discontinuous and a finite number of maxima and minima in any finite interval. (b) The signal is absolutely integrable; that is, x t dt a ( )  −   

2.1 The Continuous-Time fourier transform Energy density spectrum The total energy x of a finite-energy continuous-time complex signal Xa(t) is given by 2 dt The energy can also be expressed in terms of the CtFT a(j9) ∫)ad=n X ( is dQ2 Parseval's relation 2丌

2.1 The Continuous-Time Fourier Transform • Energy density spectrum The total energy εx of a finite-energy continuous-time complex signal xa (t) is given by ( ) ( ) ( ) 2 * x a a a  x t dt x t x t dt   − − = =   The energy can also be expressed in terms of the CTFT Xa (jΩ) ( ) ( ) 2 2 1 2 a a x t dt X j d    − −   =   Parseval’s relation

2.1 The Continuous-Time fourier transform Energy density spectrum The energy density spectrum of the continuous-time signal Xa(t is Sn(2)=|X() The energy Exr over a specified range of frequencies ≤≤ of the signal can be co omputed by over this range O Sx(@2)ds 2丌

2.1 The Continuous-Time Fourier Transform • Energy density spectrum The energy density spectrum of the continuous-time signal xa (t) is ( ) ( ) 2 xx a S X j  =  The energy x r,  over a specified range of frequencies      a b of the signal x t a ( ) can be computed by Sxx () over this range: , ( ) 1 2 b a x r xx  S d    =   

2.1 The Continuous-Time fourier transform Band-limited continuous-time signals a band-limited continuous-time signal has a spectrum that is limited to a portion of the above frequency range. An ideal band limited signal has a spectrum that is zero outside a finite frequency range C2≤|92≤9 0,0≤92<s2, However. an ideal band- limited signal cannot be <oO generated in practice

2.1 The Continuous-Time Fourier Transform • Band-limited continuous-time signals a band-limited continuous-time signal has a spectrum that is limited to a portion of the above frequency range. An ideal band￾limited signal has a spectrum that is zero outside a finite frequency range      a b ( ) 0, 0 , 0, a a b X j        =        However, an ideal band￾limited signal cannot be generated in practice