2.2 The Discrete-Time Fourier transform 02.2. 1 Definition of dtft For notational convenience, we shall use the operator symbol F to denote the DTFT of the sequence xn. Likewise, we shall use the operator symbol to denote the inverse Fourier transform. a dtFT pair will denote as x小<→X(e)
2.2 The Discrete-Time Fourier transform For notational convenience, we shall use the operator symbol F x n ( ) 1 j F X e − ⚫ 2.2.1 Definition of DTFT to denote the DTFT of the sequence x[n]. Likewise, we shall use the operator symbol to denote the inverse Fourier transform. A DTFT pair will denote as ( ) F j x n X e ⎯→
2.2 The Discrete-Time Fourier transform ●222 Properties of DTFT ◆ Periodicity property Unlike the continuous-time fourier transform it is a periodic function in w with a period 2TT. To verify this property, observe that for any integer k, Ⅹ ,八(O+2zk) ∑xn i(o+2k)n OHA-j2丌kn xinle e =X
2.2 The Discrete-Time Fourier transform ⚫ 2.2.2 Properties of DTFT ◆ Periodicity property Unlike the continuous-time Fourier transform, it is a periodic function in ω with a period 2π . To verify this property, observe that for any integer k, ( ) ( ) ( ) ( ) j k j k n 2 2 j n j kn 2 n n j X e x n e x n e e X e + − + − − =− =− = = =
2.2 The Discrete-Time Fourier transform In general, the Fourier transform x(eo)is a complex function of the real variable w and can be written in rectangular form as X(el=xr(e)+jX im/eo From above equation, it follows that xm(")=5{x(")+x(e") :e )=n{x(en)-x(")
2.2 The Discrete-Time Fourier transform In general, the Fourier transform is a complex function of the real variable ω and can be written in rectangular form as ( ) j X e ( ) ( ) ( ) j j j X e X e jX e re im = + From above equation, it follows that ( ) ( ) ( ) 1 * 2 j j j X e X e X e re = + ( ) ( ) ( ) 1 * 2 j j j X e X e X e im j = −
2.2 The Discrete-Time Fourier transform The Fourier transform can alternately be expressed in the polar from as Xe where r(el magnitude function e(o)=arg x (eio )phase function r(ejo=alejo e/[e()+2k x(elo)eo
2.2 The Discrete-Time Fourier transform The Fourier transform can alternately be expressed in the polar from as ( ) ( ) j j j ( ) X e X e e = where ( ) arg ( ) j X e = ( ) j X e magnitude function phase function ( ) ( ) ( ) ( ) j j j j k 2 j ( ) X e X e e X e e + = =
2.2 The Discrete-Time fourier transform Example 3 Real and imaginary Parts and Magnitude and phase functions of a discrete-time Fourier transform Discrete-time Fourier transform of an exponential sequence ae jo Where a=0.5
2.2 The Discrete-Time Fourier transform • Example 3 Real and imaginary Parts and Magnitude and phase functions of a discrete-time Fourier transform. Discrete-time Fourier transform of an exponential sequence ( ) 1 1 j j X e e − = − Where = 0.5