Spectral Leakage and Aliasing SPECTRAL LEAKAGE The topics in the title of this chapter are concerned with major difÞculties that are encountered in discrete signal waveform analysis and
Trang 1SINE, COSINE, AND θ 41
• To get the one-sided spectrum, we combine k = 1 with k = N − 1
and so on from 1≤ k ≤ (N /2 − 1) For k = 1 to 5, using the Mathcad
X-Y Trace tool, we get 0.3173, 0.1571, 0.1030, 0.0754, and 0.0585 The Trace tool is a very useful asset
REFERENCES
Seely, S., 1956, Radio Electronics, McGraw-Hill, New York (Also Google,
“Child-Langmuir.”)
Terman, F.E., 1943, Radio Engineer’s Handbook, McGraw-Hill, New York Zwillinger, D., ed., 1996, CRC Standard Mathematical Tables and Formulae 30th
ed., CRC Press, Boca Raton, FL
Trang 3Spectral Leakage
and Aliasing
SPECTRAL LEAKAGE
The topics in the title of this chapter are concerned with major difÞculties that are encountered in discrete signal waveform analysis and design
We will discuss how they occur and how we can deal with them The discussion still involves eternal, steady-state discrete signals
Figure 3-1a shows the “leaky” spectrum of a complex phasor using
the DFT [Eq (1-2)] at k= 7.0 (Hz, kHz, MHz, or just 7.0) whose input
signal frequency (k) may be different than 7.0 by the very small
frac-tional offset |ε| shown on the diagram For |ε| values of 10−15 to 10−6, the spectrum is essentially a “pure” tone for most practical purposes The
dots in Fig 3-1a represent the maximum spectrum attenuation at integer values of (k) for each of the offsets indicated A signal at 7.0 with the off-set indicated produces these outputs at the other exact integer (k) values Figure 3-1b repeats the example with k= 15.0, and the discrete spectrum
Discrete-Signal Analysis and Design, By William E Sabin
Copyright 2008 John Wiley & Sons, Inc.
43
Trang 444 DISCRETE-SIGNAL ANALYSIS AND DESIGN
−350
−250
−200
−150
−100
−50
0
k
(a)
(b)
10 −15
10 −6
10 −3
dB
−100
−80
−60
−40
−20
0
k dB
−300
ε =10 −3
−32
−24
−16
−8
0
k
(c)
dB
−40
N := 128 k0 := 38 n := 0,1 N−1 k := 30.0, 30.01 46 x(n) := exp n
N
j⋅2⋅π⋅ ⋅k0
X(k) : = ∑N −1 n= 0
1 N
n N
⋅ exp j⋅2⋅π⋅ ⋅(k0−k)
Figure 3-1 (a) Spectrum errors due to fractional error in frequency
speciÞcation (b) Line spectrum errors due to 10−3 fractional error in fre-quency (c) Continuous spectrum at 38 and at minor spectral leakage loops (d) Continuous spectrum of real, imaginary, and magnitude over a
nar-row frequency range (e) Time domain plot of the u(t) signal, real sine
wave plus dc component Imaginary part of sequence = 0 (f) Real and
imaginary components of the spectrum U(k) of time domain signal u(t).
(g) Incorrect way to reconstruct a sine wave that uses fractional values of time and frequency increments
Trang 5SPECTRAL LEAKAGE AND ALIASING 45
0 20 40 60 80 100 120
−1
0
1
2
Re(u(t))
Im(u(t))
t
0 20 40 60 80 100 120
−1
−0.5
0
0.5
Im(U(k))
Re(U(k))
k
z := 0, 0.5 N −1
Q := 0, 0.5 N −1
v(z) :=
0 20 40 60 80 100 120
−1
0
1
2
Re(v(z))
Im(v(z))
z
U(k) : = 1 ∑N−1t= 0
N
t N u(t) ⋅exp −j⋅2⋅π⋅k⋅
u(t) : = 0.5 + sin 2 ⋅π⋅2⋅N t
N : = 2 7 t : = 0,1 N−1 k : = 0,1 N−1
⋅
∑N−1
Q = 0 (U(Q)) ⋅ exp j⋅2⋅π⋅Q⋅ N z
(d )
(e)
(f )
(g)
−40
−32
−24
−16
−8
0
Mag
Figure 3-1 (continued)