SINE, 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. 3 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 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 44 DISCRETE-SIGNAL ANALYSIS AND DESIGN 0123456789101112131415 −350 −250 −200 −150 −100 −50 0 k (a) (b) 10 −15 10 −6 10 −3 dB 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 −100 −80 −60 −40 −20 0 k dB −300 ε =10 −3 30 32 34 36 38 40 42 44 4 6 − 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. SPECTRAL 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) : = ∑ N−1 t = 0 1 N t N u(t)⋅exp −j⋅2⋅π⋅k⋅ u(t) : = 0.5 + sin t N 2⋅π⋅2⋅ N : = 2 7 t : = 0,1 N−1 k : = 0,1 N−1 ⋅ ∑ N−1 Q = 0 (U(Q)) ⋅ exp j⋅2⋅π⋅Q⋅ z N (d ) (e) (f ) (g) −40 −32 −24 −16 −8 0 dB Imag Real Mag 37 38 37.5 38.5 39 Figure 3-1 (continued) . 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 44 DISCRETE-SIGNAL ANALYSIS AND DESIGN 0123456789101112131415 −350 −250 −200 −150 −100 −50 0 k (a) (b) 10 −15 10 −6 10 −3 dB 0. 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. 3 Spectral Leakage and Aliasing SPECTRAL. 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,