DISCRETE-SIGNAL ANALYSIS AND DESIGN- P15 doc
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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.
