DISCRETE-SIGNAL ANALYSIS AND DESIGN- P15 doc

56 DISCRETE-SIGNAL ANALYSIS AND DESIGN over the region 0 to 6000 Hz between adjacent channels. In a 1.0-Hz bandwidth, P(f) 1Hz BW dBm =−40 dBm − 20  f 6000  dB (3-6) We convert dBm per Hz to watts per Hz, integrate from 0 to 6000 Hz, and adjust to the 300-Hz instrument bandwidth: P out of band (W) = 300 6000  6000 0 P(f)df = 6.45 × 10 −6 W in 300 Hz (3-7) The average P out of band in dBm in a 300-Hz band is −21.90 dBm. The ratio of in-band to out-of-band power for a 300-Hz bandwidth is 24.77 dBm −(−21.90 dBm) =46.7 dB. Note the method of employing dBm and dB in an equation. If both sides of the Þlter are considered (which they are not at this time), the ratio becomes 3 dB smaller. We want the power in the alias zone shown in Fig. 3-6. To get this, inte- grate the out-of-band spectrum P(f ) from 3000 to 6000 Hz, multiply by 2 to get both halves of the alias zone, and adjust for the 300-Hz instrument bandwidth: P alias = 2  300 3000   6000 3000 P(f) df = 2.345 × 10 −6 W =−26.3 dBm (3-8) The ratio of in-band power to alias band power between the two bands shown is 24.77 dBm −(−26.3 dBm) =51.07 dB. Subtract 3 dB for an addi- tional alias band on the left side of the diagram. The instrument for this measurement can be a calibrated spectrum analyzer. Since the noise signal is the same kind at every frequency and amplitude of interest in this example, we do not worry about the fact that the amplitude reading for noise on a spectrum analyzer is not quite the same as for a sine wave. The relative dB readings are cor- rect. Also, in Eqs. (3-7) and (3-8) we are Þnding the average power over the speciÞed band and then normalizing that average power to a SPECTRAL LEAKAGE AND ALIASING 57 (a) ( b ) 1.051 dB 0 300 600 900 Detail of 300 Hz steps 10 kHz 10 kHz 6 kHz 6 kHz 300 Hz 300 Hz 300 Hz Ppb = 300 mW (24.8 dBm) in 300 Hz 0 dBm (1mW) in 1 Hz BW −40 dBm −60 dBm Alias Zone 10 W in-band Figure 3-6 Power in the aliasing zone. 300 Hz bandwidth. It is important to not change any analyzer settings that might affect the internal adjustments or calibrations of the instru- ment. Finally, if the spectrum analyzer contains a high-quality tracking generator, it can be used as a sine-wave signal source instead of a noise generator. Figure 3-7 is the Mathcad worksheet used to perform the cal- culations for this example which could be a template reference for the procedure. In more complicated (irregular) examples it may be necessary to divide the various frequency ranges into narrow non-overlapping frequency strips, to analyze each strip individually, and to combine the results in a manner similar to that suggested here. In wideband measurements it is often necessary to verify the instrument calibrations across the measure- ment frequency range and to eliminate spurious system responses that can invalidate the results. 58 DISCRETE-SIGNAL ANALYSIS AND DESIGN Ppb := 300 10000 0 10000 .001df × Ppb = 0.3 watts in 300 Hz pass band PdBm = 24.7712 dBm in 300 Hz pass band out-of-band frequency index PdBm := 10 × log Ppb .001 f := 0,1 6000 dBm(f) := −40 −20 × f 6000 P(f) := 0.001 × 10 dBm(f) 10 Poutb := 300 6000 0 6000 P(f) df × Poutb = 6.4493 × 10 −6 Poutb .001 PoutdBm := 10 × log PoutdBm = −21.9049 10 × log 2 × Poutb Ppb =−44 dB for sum of both out-of-band regions Pazone := 2 × 300 3000 3000 6000 P(f) df watts in alias zone in 300 Hz band Pazone := 2.345 × 10 −6 × watts in alias zone in 300 Hz band 10 × log Pazone Ppb = −51.1 dB below passband Figure 3-7 Calculation of power in the alias zone of Fig. 3-6. SPECTRAL LEAKAGE AND ALIASING 59 ALIASING IN THE TIME DOMAIN Aliasing has been considered primarily in the X (k ) frequency domain, where bandlimited spectra overlap. But aliasing also occurs in the time domain, where periodic x(n) time sequences similar in appearance to Figs. 3-3 and 3-4 overlap or are truncated or interrupted prematurely before the sample values become insigniÞcant. An oscilloscope can easily show the overlap between two separate and independent time-sequence generators that are triggered alternately; the two could be triangular waves. After the DFT transformation the result is often an unacceptable modiÞ- cation of the spectrum. It is important that all of the signiÞcant data in the time-domain data record be obtained and utilized and that this record has sufÞcient resolution to include both high-frequency and low-frequency elements. Some smoothing or windowing of the time-domain waveform prior to the Fourier transformation may be desirable to reduce spurious high-frequency irregularities that might mask important results. These sub- jects are described in greater detail in Chapter 4. REFERENCES Sabin, W. E., 1995, The lumped element directional coupler QEX (ARRL), March. Carlson, A. Bruce, 1986, Communication Systems, 3rd ed., McGraw-Hill, New York. . to 6000 Hz, and adjust to the 300-Hz instrument bandwidth: P out of band (W) = 300 6000  6000 0 P(f)df = 6.45 × 10 −6 W in 300 Hz (3-7) The average P out of band in dBm in a 300-Hz band is −21.90. range and to eliminate spurious system responses that can invalidate the results. 58 DISCRETE-SIGNAL ANALYSIS AND DESIGN Ppb := 300 10000 0 10000 .001df × Ppb = 0.3 watts in 300 Hz pass band PdBm. band is −21.90 dBm. The ratio of in-band to out-of-band power for a 300-Hz bandwidth is 24.77 dBm −(−21.90 dBm) =46.7 dB. Note the method of employing dBm and dB in an equation. If both sides