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RTSA terminology
What is real time
Real-time digital signal processing (DSP):
The term “real time” is derived from early work on digital simulations of
physical systems. A digital system simulation is said to operate in real
time if its operating speed matches that of the real system which it is
simulating.
To analyze signals in real time means that the analysis operations must
be performed fast enough to accurately process all signal
components in the frequency band of interest. This definition implies
that we must:
1. Sample the input signal fast enough to satisfy Nyquist criteria. This
means that the sampling frequency must exceed twice the bandwidth
of interest.
2. Perform all computations continuously at a fast enough rate that the
output analysis keeps up with the changes in the input signal.
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April 13, 2010
Two important needs for Real Time Spectrum Analyzers
ƒ Processing all information
contained in a signal in real
time requires:
–
–
–
–
–
Enough capture bandwidth to
support the signal of interest.
High enough ADC clock rate to
exceed the Nyquist criteria for
the capture bandwidth.
Long enough capture interval to
support the narrowest resolution
bandwidth (RBW) of interest.
High enough DFT transform rate
to exceed the Nyquist criteria for
the RBW of interest.
Overlapping DFT frames
•
•
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April 13, 2010
The amount of overlap depends on
the window function
The Window function is determined
by the RBW
ƒ
Discovering, and Capturing
transient events requires:
– Minimum event duration for 100%
probability of capturing a single
non-repetitive event.
• A minimum event is defined as the
narrowest rectangular pulse that can
be captured with certainty.
– Enough capture bandwidth to
support the signal of interest.
– High enough ADC clock rate to
exceed the Nyquist criteria for the
capture bandwidth.
– Long enough capture interval to
support the narrowest resolution
bandwidth (RBW) of interest.
– High enough DFT transform rate
to support the minimum event
duration.
Performing repetitive Discrete Fourier Transforms is
equivalent to passing signals through a bank-of-filters
DFT* Based Spectrum Analysis
Input signal
Memory Contents
A/D
DFT Engine
time
Time samples
N-point FFT
Ti m
e
Equivalent bank of filters
Bank of N Bandpass
filters with centers
separated by one FFT
frequency bin width
Input signal
time
Complex
Envelope
detection
M/θ
M/θ
M/θ
Sampled at the
rate at which
transforms are
computed
M/θ
* The Fast Fourier Transform (FFT) is a common implementation
of a Discrete Fourier Transform (DFT).
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April 13, 2010
Real-time spectrum analysis
ƒ
Discrete Fourier transforms
–
ƒ
Spectrum analysis, also called Fourier analysis, requires that signals be observed
in the frequency domain. When using DSP, this implies performing discrete
Fourier transforms (DFTs) on time sampled data.
Real time Fourier analysis using DSP
–
–
–
–
Refer to diagram on the previous slide
Discrete Fourier transforms are continuously performed on the sampled input
signal.
This is equivalent to passing the signal through a bank of bandpass filters, each
of which has the bandwidth and separation of the DFT bins. The complex
envelope (I and Q or Magnitude and Phase) is computed for each frequency
domain bin of the DFT output each time a new DFT is performed.
Criteria for Real Time Spectrum analysis
1.
2.
The input signal must be sampled fast enough to meet the Nyquist criteria for the
bandwidth of interest.
The DFT computations must be performed fast enough such that the Nyquist
criteria is met for each of the DFT bins.
•
•
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April 13, 2010
It can be shown that this is equivalent to having no gaps between DFT frames.
Windowing and other practical implementations require DFT frames to overlap.
Windowing and DFT frame overlap
Processing
gap between
DFT frames
Window
1.2
1
0.8
0.6
0.4
0.2
0
Window
1.2
1
0.8
0.6
0.4
0.2
0
1
1
7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97
103
109 115
7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97
103
109 115
121
127
DFT Frames
end-to-end
with no gap
0.6
0.4
0.2
0
1
2
2
2
1.5
1.5
1
1
1
0.5
0.5
0.5
0
0
0
133
183
233
283
333
383
-17
-0.5
33
83
133
183
233
283
333
383
-17
-0.5
-1
-1
-1
-1.5
-1.5
-1.5
-2
-2
-2
2
2
2
1.5
1.5
1
1
1
0.5
0.5
0.5
83
133
183
233
283
333
383
0
-17
-0.5
33
83
133
183
233
283
333
383
0
-17
-0.5
-1
-1
-1
-1.5
-1.5
-1.5
-2
-2
-2
Frame 2 Windowed signal
2
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
83
133
183
233
283
333
383
0
-17
-0.5
33
83
133
183
233
283
333
383
0
-17
-0.5
-1
-1
-1
-1.5
-1.5
-1.5
-2
-2
-2
Frame 3 Windowed signal
2
2
1.5
1.5
1
1
1
0.5
0.5
0.5
83
133
183
233
283
333
383
0
-17
-0.5
33
83
133
183
233
283
333
383
0
-17
-0.5
-1
-1
-1
-1.5
-1.5
-1.5
-2
-2
-2
April 13, 2010
55
61
67
73
79
85
91
97
103
109 115
121
127
33
83
133
183
233
283
333
383
33
83
133
183
233
283
333
383
33
83
133
183
233
283
333
383
283
333
383
Frame 3 Windowed signal
1.5
33
49
Frame 3 Windowed signal
2
0
-17
-0.5
43
Frame 2 Windowed signal
2
33
37
Frame 2 Windowed signal
2
0
-17
-0.5
31
Frame 1 Windowed signal
1.5
33
25
Frame 1 Windowed signal
Frame 1 Windowed signal
0
-17
-0.5
19
Input Signal
1.5
83
13
2.5
2.5
33
7
Input Signal
Input Signal
-17
-0.5
Overlapping
DFT Frames
1
0.8
121 127
2.5
5
Window
1.2
33
83
133
183
233
Implications of overlapping DFT frames
Non-Overlapping DFTs
Adjacent DFT Frames with no gap
Continuously Sampled Input
Continuously Sampled Input
Data is lost
Input Time
samples
Con
sam tinuou
s
freq ples a Outpu
t ea
u en
t
cy B ch
in
Input Time
samples
Co
sa ntin
fre mp uo
qu les us
e n at O u
c y ea t p
B i c h ut
n
Overlapping DFTs
Continuously Sampled Input
•Non-overlapping DFT : Not Real Time
Input Time
samples
Co
sa ntin
fre mp uo
qu les us
e n at O u
c y ea t p
B i c h ut
n
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April 13, 2010
•Adjacent DFT frames with no gap: Minimum
needed to meet real time criteria for unwindowed (rectangular window) FFT processing
•Overlapping DFT frames: Needed to meet real
time criteria for windowed DFT processing. The
amount of overlap must increase as the window
narrows.
Minimum Event Duration for 100% probability of
discovery or capture at full amplitude
Where:
Overlapped DFT frames
Tmin=2 Tacq- Tol
Tmin=Minimum time for 100% probability if
intercept
Gap between frames
Tol=Overlap time
Tmin=2 Tacq+ Tgap
Tgap=Gap between acquisitions when there is
no overlap
Tacq=Time to acquire a DFT frame
Overlapped DFT frames
Gap between DFT frames
Minimum pulse duration must
contain at least one full acquisition
Acquire data
Minimum pulse duration must
contain at least one full acquisition
Compute DFT
Acquire data
Acquire data
Compute DFT
Acquire data
Compute DFT
Acquire data
Compute DFT
Compute DFT
Gap
Overlap
Time
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April 13, 2010
Time
Minimum Event Duration for 100% Probability of
Intercept and DFT overlap.
Model
Max
Acquisition
Bandwidth
DPX
Frequency Mask Trigger
Power/Level
Trigger
Min Pulse Duration
for 100%
probability of
discovery
DFT overlap
Min Pulse duration
for 100%
probability of
trigger
DFT overlap
Minimum pulse
duration for 100%
probability of trigger
at widest BW
RSA6000
40 MHz
31 µSec
≧50 % for
spans ≦10 MHz
30.1 µSec
≧50 %
20 nSec
RSA6000
110 MHz
24 µSec
≧50 % for
spans ≦ 10
MHz
10.24 µSec
≧50 %
6.67 nSec
RSA 3408B
36 MHz
31 µSec
≧50 % for Span
≦ 10 MHz
20 µSec
≧50 %
20 nSec
RSA 3300B
15 MHz
41 µSec
≧50 % for
spans ≦ 10
MHz
30 µSec
≧50 %
40 nSec
Opt 110
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April 13, 2010
Appendix
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April 13, 2010
Sampling rate vs DFT rate in RTSAs
Meeting Nyquist criteria for sampled systems
– Real-valued input signal input:
•
The rate at which time domain samples are taken is at least twice the
bandwidth of the signal of interest.
– Complex valued signal input (I and Q):
•
The rate at which I and Q are each sampled must be at least equal to the
bandwidth of the signal of interest. Each complex input is effectively two
samples.
– Complex valued output
•
•
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April 13, 2010
Complex output is the general case for DFT computations. Either the
Cartesian (I and Q) or the polar ( magnitude and phase) are commonly used.
The rate at which new DFTs are computed is at least equal to the bandwidth
of each output bin. Each complex output for each DFT bin is effectively two
samples.
The need for DFT overlap.
Non-Overlapping DFTs
N= number of points in the FFT
Continuously Sampled Input
G
G= number of samples in the gap
FS=Sampling frequency
Input Time
samples
N
R=Transform rate
BW= output bin width
Con
sam tinuou
s
freq ples a Outpu
t ea
u en
t
cy B ch
in
BW=FS/N
R=FS/(N+G)
R<BW for G>0
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April 13, 2010
ƒ
Meeting the Nyquist criteria for
complex valued samples requires
that the transform rate, R be at
least equal to the bin width, BW.
ƒ
There must be overlap
The effect of Windowing
ƒ The mathematics of finite length Discrete Fourier transforms
have the inherent assumption that the signal is periodic, with a
period equal to the length of the transform.
ƒ There can be edge effects that cause spectral leakage when
the time-domain signal in the DFT does not have this exact
periodicity.
ƒ Windowing functions are used to de-emphasize the samples in
the edges of a DFT frame. The window function is multiplied
with the input samples on a sample-by-sample basis.
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April 13, 2010
Some definitions from a Google search
ƒ Real-time systems are defined as those systems in which the
correctness of the system depends not only on the logical result
of computations, but also on the time at which the results are
produced.
– Real-Time Systems
•
•
The International Journal of Time-Critical Computing Systems
Editor-in-Chief: Tarek F. Abdelzaher; Giorgio Buttazzo; Krithi Ramamritham
ƒ A computer system that responds to input signals fast enough
to keep an operation moving at its required speed. Examples
are video game computers and videoconferencing systems as
well as computers used to control airplanes, space shuttles and
other "real" equipment.
– PCMAG.com encyclopaedia
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April 13, 2010