83.1 Discrete-Time Fourier Transform Unless otherwise stated, we shall assume that the phase function ((o) is restricted to the following range of values π≤θ(0)≤ called the principal value
§3.1 Discrete-Time Fourier Transform • Unless otherwise stated, we shall assume that the phase function () is restricted to the following range of values: - () called the principal value
83.1 Discrete-Time Fourier Transform The DTFTs of some sequences exhibit discontinuities of 2t in their phase responses An alternate type of phase function that is a continuous function of o is often used It is derived from the original phase function by removing the discontinuities of2π
§3.1 Discrete-Time Fourier Transform • The DTFTs of some sequences exhibit discontinuities of 2 in their phase responses • An alternate type of phase function that is a continuous function of is often used • It is derived from the original phase function by removing the discontinuities of 2
83.1 Discrete-Time Fourier Transform Example-The dtFT of the unit sample sequence 8n is given by △(e)=∑8[neon=8O]=1 Example- Consider the causal sequence x[n]=a"u[n]a<1
§3.1 Discrete-Time Fourier Transform • Example - The DTFT of the unit sample sequence d[n] is given by ( ) = d[ ] = d[0] =1 − =− j n n j e n e x[n] = [n], 1 n • Example - Consider the causal sequence
83.1 Discrete-Time Fourier Transform · Its DTFT is given by 1= n=o e on X(e/0)=∑ un]e j=∑a ∑(oe-)=.1 0 1-ae o ae Jo=a <1
§3.1 Discrete-Time Fourier Transform • Its DTFT is given by = = = − =− − 0 ( ) [ ] n n j n n j n j n X e n e e − − = − = = j e n j n e 1 1 0 ( ) = 1 − j as e
83.1 Discrete-Time Fourier Transform The magnitude and phase of the dtft ⅹ(e°) 1/(1-0.5e-Jo)are shown below 04 0.4
§3.1 Discrete-Time Fourier Transform • The magnitude and phase of the DTFT X(ej) = 1/(1 – 0.5e-j) are shown below