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）

2.1 The Continuous-Time Fourier transform Band-limited continuous-time signals Band-limited signals are classified according to the frequency rang where most of the signal's energy is concentrated lowpass continuous-time signal highpass continuous-time signal bandpass continuous-time signal

2.1 The Continuous-Time Fourier Transform • Band-limited continuous-time signals Band-limited signals are classified according to the frequency rang where most of the signal’s energy is concentrated. lowpass continuous-time signal highpass continuous-time signal bandpass continuous-time signal

2.2 The Discrete-Time Fourier transform 02.2. 1 Definition of dtft The discrete-time Fourier transform(DTFT)X(e)of a sequence xn] is defined by X(em)=∑xem As can be seen from the definition, The discrete-time Fourier transform(DTFT) X(e)of a sequence xnj is a function of the normalized angular frequency

2.2 The Discrete-Time Fourier transform ⚫ 2.2.1 Definition of DTFT The discrete-time Fourier transform (DTFT) of a sequence x[n] is defined by ( ) j X e  ( )   j j n n X e x n e    − =− =  As can be seen from the definition, The discrete-time Fourier transform (DTFT) of a sequence x[n] is a function of the normalized angular frequency. ( ) j X e 

2.2 The Discrete-Time Fourier transform 02.2.1 Definition of dtft Example 1 Discrete-time Fourier transform of the unit sample sequence X(e")=∑b小le Jon

2.2 The Discrete-Time Fourier transform ⚫ 2.2.1 Definition of DTFT Example 1 Discrete-time Fourier transform of the unit sample sequence ( )   1 j j n n X e n e     − =− = = 

2.2 The Discrete-Time Fourier transform 02.2. 1 Definition of dtft Example 2 Discrete-time Fourier transform of an exponential sequence X(e")=∑a[小lem=∑ n- on a e ae

2.2 The Discrete-Time Fourier transform ⚫ 2.2.1 Definition of DTFT Example 2 Discrete-time Fourier transform of an exponential sequence ( )   1 1 j n j n n j n j n n X e u n e e e          − − − =− = = = −  

2.2 The Discrete-Time Fourier transform 02.2. 1 Definition of dtft The Fourier coefficients xn] can be computed from x(elo) using the Fourier integral given by [n]=o x(ele inverse discrete-time Fourier transform

2.2 The Discrete-Time Fourier transform The Fourier coefficients x[n] can be computed from using the Fourier integral given by ( ) j X e    ( ) 1 2 j j n x n X e e d       − =  inverse discrete-time Fourier transform ⚫ 2.2.1 Definition of DTFT