Polar Representation and Graphical Plot of DTFT
3.3 Polar Representation and Graphical Plot of DTFT
Similarly to the continuous-time case, we can write the polar representation of the DTFT as
X (Ω) = |X(Ω)|∠X(Ω)
If x[n] is real, then its DTFT X (Ω) = F{x[n]} has the following properties: - X (Ω) is periodic with period 2π in Ω.
- The magnitude |X(Ω)| is an even function of Ω and is symmetric about Ω = mπ (m: an integer).
- The phase ∠X (Ω) is an odd function of Ω and is anti-symmetric about Ω = mπ (m: an integer).
Note that all the information about the DTFT of a real-valued sequence is contained in the frequency range [0, π ] since the portion for other ranges can be determined from that for [0, π ] using symmetry and periodicity. Consequently, we usually plot the spectrum for 0 ≤ Ω ≤ π only.
Remark 3.3 Phase Jumps in the DTFT Phase Spectrum From the phase spectra shown in Fig. 3.1(c1)–(c2) and 3.7(c1)–(c2), it can be observed that there are two occasions for which the phase spectrum has discontinu- ities or jumps:
- A jump of ±2π occurs to maintain the phase value within the principal range of [−π, +π]. - A jump of ±π occurs when the sign of X(Ω) changes.
3.3 Polar Representation and Graphical Plot of DTFT 145 The sign of phase jump is chosen in such a way that the resulting phase spectrum
is odd or antisymmetric and lies in the principal range [−π, +π] after the jump.
Remark 3.4 The DTFT Magnitude/Phase Spectra of Symmetric Sequences (1) Especially for anti-symmetric sequences, their magnitude spectrum is zero at
Ω = 0 (see Fig. 3.7(b1)–(b2)). This implies that the DC gain of digital filters having an anti-symmetric impulse response is zero so that they cannot be used as a lowpass filter.
(2) As for the sequences that are even/odd about some point, their DTFTs have linear phase −MΩ (proportional to the frequency) except for the phase jumps
so that the DTFT phase spectra are piecewise linear. Also, symmetric/anti- symmetric sequences, that are just shifted versions of even/odd ones, preserve linear phase because shifting does not impair the linear phase characteristic (see Eq. (3.2.6)). This is illustrated by Examples 3.1, 3.11, and 3.12 and will be restated in Remark 4.8, Sect. 4.6.
Example 3.11 DTFT of an Odd Sequence For an odd sequence
n=
x 1 [n] = · · · 0 1 2 1 0 −1 −2 −1 0 0 ··· (E3.11.1)
2 2 x 1 (t )
x 1 [n ] = x 1 (n T ) | T = 1 x 2 [n ]
0 t –5
|| T 0 5 5 1 3 –1 –2
–2 (a1) An odd sequence
(a2) An anti–symmetric sequence
DTFT 6 CTFT 6
| X 1 (Ω) |
| X 1 ( ω ) /T
| X | (Ω) 2 |
(b2) The DTFT magnitude spectrum of x 2 [n] ∠ X (Ω)
(b1) The CTFT/DTFT magnitude spectra of x 1 (t ) /x 1 [n]
(c2) The DTFT phase spectrum of x 2 [n] Fig. 3.7 The DTFT spectra of an odd sequence and an anti-symmetric sequence
(c1) The DTFT phase spectrum of x 1 [n]
146 3 Discrete-Time Fourier Analysis we have its DTFT as
1 (Ω) = x 1 [n] e − j Ωn
n=−∞ j 3Ω
−jΩ =e (D.22) −e + 2(e −e )+e −e = j2 sin(3Ω)
j π/ + j4 sin(2Ω) + j2 sin(Ω) 2 = 4 sin(2Ω) (1 + cos Ω) e (E3.11.2)
(D.12)
Noting that the continuous-time version of this sequence is apparently the sum of two opposite-sign shifted triangular pulses, we can compute its CTFT spectrum as
x (E2.3.4) with A=2,D=2
1 (t ) = 2(λ D (t + 2) − λ D (t − 2)) (2.5.6) with t ↔
) = j8 sin(2ω) sinc π
This sequence and its DTFT magnitude/phase spectra are depicted in Fig. 3.7(a1) and (b1)/(c1), respectively. The CTFT magnitude spectrum (E3.11.3) (divided by sampling interval or period T ) is plotted (in a dotted line) together with the DTFT magnitude spectrum in Fig. 3.7(b1) for comparison.
Example 3.12 DTFT of an Anti-Symmetric Sequence For the sequence which is anti-symmetric about n = 1.5
we have its DTFT as
X 2 (Ω) = x 2 [n] e − j Ωn = −1 + 2 e −jΩ − 2e − j2Ω +e − j3Ω
n=−∞
= −(1 − e − j3Ω ) + 2(e −jΩ −e − j2Ω ) − j3Ω/2 (e j 3Ω/2
j Ω/ = −e 2 −e − j3Ω/2 ) + 2e − j3Ω/2 (e −e − j Ω/2 ) (D.22) =2je − j3Ω/2
2 2 −3 2 + 2 This sequence and its magnitude/phase spectra are depicted in Fig. 3.7(a2) and (b2)/
(c2), respectively.
3.4 Discrete Fourier Transform (DFT) 147