If both sides of the Þlter are considered which they are not at this time, the ratio becomes 3 dB smaller.. The instrument for this measurement can be a calibrated spectrum analyzer.. Si
Trang 1over 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
We convert dBm per Hz to watts per Hz, integrate from 0 to 6000 Hz, and adjust to the 300-Hz instrument bandwidth:
Pout of band(W)= 6000300
6000
0 P (f ) df = 6.45 × 10−6W in 300 Hz
(3-7)
The average Pout 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:
Palias = 2
300 3000
6000
3000 P (f ) df
= 2.345 × 10−6W= −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
Trang 2SPECTRAL LEAKAGE AND ALIASING 57
(a)
(b)
1.051 dB
0 300 600 900
Detail of 300 Hz steps
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
Trang 3Ppb := 300
10000 001df
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 P(f) df
×
.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
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
Figure 3-7 Calculation of power in the alias zone of Fig 3-6.
Trang 4SPECTRAL 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