# Digital Signal Processing _in 2700 seconds_ by bestt571

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```									                            Digital Signal Processing
(in 2700 seconds)

John Carwardine and Frank Lenkszus

Digital Signal Processing in 2700 seconds
Some Applications of Digital Signal Processing

Consumer Applications            Accelerator Applications
• Communications                 • BPM Processing
– Digital cellular phones    • Feedback control
– Echo cancellation              – orbit control
• Data compression                   – multi-bunch feedback
– HDTV                       • RF applications
– MP3 digital audio              – direct digital down-conversion
• Video games                        – digital I/Q sampling
• Automobiles                    • Accelerator tuning
– Engine management              – spectral estimation
– Adaptive suspension        • High precision power supplies

Digital Signal Processing in 2700 seconds
Overview

•    DSP Essentials
•    Filtering BPM data
•    Direct digital down-conversion
•    Optimal & Adaptive Filters
•    DSP Hardware

Digital Signal Processing in 2700 seconds
“Black Box” View

Signal
Processor

Analog                      Analog
Filter                      Filter

Digital Signal Processing in 2700 seconds
Key DSP Operations

•   Filtering                               •    Transformation into another domain

•   Modulation and demodulation             •    Correlation of two signals

•   Signal generation, frequency synthesis

D/A
Processor
Converter

Digital Signal Processing in 2700 seconds
Discrete-Time vs Continuous-Time

•    Continuous-time signals are functions of a continuous-valued independent
variable t, and they exist at all values of t.

xa(t)                                                    t
0

•    Discrete-time signals are functions of an integer-valued index (eg n, m, k),
and the signals have no meaning for non-integer values of the independent
variable.
x[-3]
x[1]

2 3 4
x[n]                                                    n
-4 -3 -2 -1 0 1

Digital Signal Processing in 2700 seconds
Ambiguity of Sampled-Data Signals

•    Which continuous-time signal does this discrete-time sequence represent?

•    Knowing the sampling rate, is not enough to uniquely reconstruct a
continuous-time signal from a discrete-time sequence.
•    The uncertainty is a result of aliasing.

Digital Signal Processing in 2700 seconds
Aliasing of Tones
•    Single Frequency Tones greater than fs/2 appear as aliases.
•    Consider the following spectrum that is sampled at 1600Hz.

1100 Hz         1800 Hz

200 Hz               500 Hz

fs                   fs              3 fs     2 fs
2                                     2
3200 Hz
800 Hz              1600 Hz

Digital Signal Processing in 2700 seconds
How to Avoid Aliasing

•    Digitize the analog signal at least 2x the highest frequency component
(Shannon’s Sampling Theory).
•    Use analog anti-aliasing filter to get rid of high frequency components before
the digitizer.

•    Realize there will always be aliasing to some degree, the question is how much
can be tolerated...

Digital Signal Processing in 2700 seconds
APS Turn-by-Turn Beam Position Monitors

•    360 beam position monitors in each plane.
•    BPMs are digitized every 7.4µS
•    Data is averaged to get rid of high frequency noise so it
can be used for orbit control
– RT orbit feedback, running at 1.6KHz
– Orbit Correction, running at 0.5Hz

Digital Signal Processing in 2700 seconds
BPM Memory Scanner
•    Boxcar averaging is used to lowpass filter turn-by-turn data for the orbit
correction systems.

2048
slow orbit
135KHz                      point
correction
0.5Hz
averager

BPM

32       real time
point     feedback    1.6KHz
averager     system

Digital Signal Processing in 2700 seconds
Averager Block Diagram (DSP Viewpoint)
x[n-1]              x[n-2]              x[n-3]
1-sample            1-sample            1-sample
x[n]              delay               delay               delay

Σ
1/4
y[n]
•    This can be described with the following difference equation

y[n ] = 0.25 ⋅ ( x[n ] + x[n − 1] + x[n − 2] + x[n − 3])

•    Or with the following z-transform transfer function
Y ( z) 1
H lp ( z ) =       = (1 + z −1 + z − 2 + z −3 )
X ( z) 4

Digital Signal Processing in 2700 seconds
Averagers with Different Number of Points

0

-5
Attenuation (dB)

-10

-15

-20
4-point
-25      16-point
32-point
-30 -3             -2          -1          0
10          10          10          10
Normalized Frequency

Digital Signal Processing in 2700 seconds
32-Tap Averager vs 32-Tap FIR Filter

•                      A boxcar averager is simple to implement, but does not provide the optimum
level of filtering

Averager Coefficients
0
Attenuation (dB)

-10

-20

-30                                             FIR Filter Coefficients
-40
Averager
-50         FIR Filter
-3             -2        -1          0
10             10        10          10
Normalized Frequency

Digital Signal Processing in 2700 seconds
Ideal Frequency-Selective Digital Filters
•    An ideal frequency-selective lowpass filter has a passband with constant
magnitude, an infinitely sharp transition between passband and stopband, and
infinite attenuation in the stopband. The phase delay is zero for all frequencies.
A(f)
1       passband

stopband
f
0              Fc
0.5Fs
Fs-Fc
Fs

•    The impulse response (coefficient weights) of this ideal filter follow a
doubly-infinite sin(x)/x function

n
0

Digital Signal Processing in 2700 seconds
Radio Frequency Applications of DSP

•    Two particular methods of sampling RF signals are gaining attention because of
the advent of high-speed A/D converters.
•    Both are associated with sampling band-limited signals that ride on a high-
frequency carrier, eg
– sampling RF probes for cavity field control.
– accelerator tune measurement.

•    Direct digital down-conversion (software radio)
– eliminates the need for a conventional analog RF mixer.

•    Digital I/Q sampling
– eliminates difficulties associated with detecting in-phase and quadrature
components of an RF signal.

Digital Signal Processing in 2700 seconds
Sampling Band-limited Signals
•    Consider a 2MHz band-limited signal riding on an 8MHz carrier.

f (MHz)
-9    -7             0                7      9

•    The IF could be extracted by mixing with a local oscillator at 10MHz and
sampled at 6MHz, or could be directly sampled at > 18MHz.

1-3MHz
I.F.
Signal Input
(7-9MHz)
A/D

Fs > 6MHz
Local oscillator
(10MHz)

Digital Signal Processing in 2700 seconds
Bandpass Sampling Example
•    Instead, let’s directly sample the signal at only 10M samples/second.

Image               Original
spectrum            spectrum

f (MHz)
-10              -5       0   1     3      5      7      9     10
(Fs/2)               (Fs)

•    In this case the Nyquist frequency would be 5MHz, and the original
spectrum is in the range of Fs/2 to Fs, instead of the range DC-Fs/2 (as we
are used to seeing).
•    The original spectrum is aliased into the lower half of the frequency band,
reflected about the Nyquist rate of 5MHz, appearing in the frequency
range 3Mhz - 1MHz.
•    So, we have successfully sampled the signal using a sampling rate almost
half the ‘officially’ required rate

Digital Signal Processing in 2700 seconds
Bandpass Sampling Example (cont)
•    What if we sample at only 6.5M samples/second??

Image             Original
spectra          spectrum

0.5   2.5    4   6                    f (MHz)
3.25         6.5      9.75
(Fs/2)       (Fs)

•    This time the original spectrum lies between Fs and 1.5Fs.
•    Here, the spectrum is reflected about the sampling rate, to appear in the
range from Fs/2 to Fs, spanning 6MHz - 4MHz.
•    It is then reflected a second time about Fs/2, finally appearing in the lower
half of the sampled frequency range between 0.5MHz and 2.5MHz.

Can we sample at an even lower rate and still get a unique spectrum??

Digital Signal Processing in 2700 seconds
General Case of Bandpass Sampling
•    In general, it can be shown that if there are m image spectra between the original
and its negative image, the range of possible sampling frequencies is given by
the expression

2 fc − B        2 fc + B
≥ fs ≥
m             m +1

•    Example with m = 5

2fc+B

2fc-B

f
-3fs -fc   -2fs          -fs         0         fs   2fs   fc 3fs

Digital Signal Processing in 2700 seconds
Analog I/Q Detector

Baseband
Signal Input
A/D        I
(476MHz)
sin

Baseband
Local oscillator                  0o
(476MHz)
A/D        Q
-90o
cos

•    Issues: DC offsets in mixer, quadrature phase errors, impedance matching, ...

Digital Signal Processing in 2700 seconds
Quadrature Sampling with Digital Mixing

•    Digital technology now offers a completely digital approach to this problem.

I

4.9MHz                        Digital
I.F.             sin(wn)   lowpass
Signal Input
(476MHz)
A/D

19.6MHz                          Q
Local oscillator
(471.1MHz)
Digital
cos(wn)   lowpass

•      The continuous-time signal is sampled at exactly 4 times the IF frequency.
•      Digital sine and cosine signals are multiplied with the incoming discrete-
time sequence to generate the real and imaginary part of the signal.

Digital Signal Processing in 2700 seconds
Optimal and Adaptive Filters
•    Consider a situation where a signal x[n] is to be filtered so that the output
sequence is as close as possible to a desired signal d[n]

d[n]
e[ n] = d [ n] − ∑ f [k ] ⋅ x[n − k ]
+
y[n]                                                    k
x[n]            f[n]                    Σ              e[n]
-

•    If the statistics of the input process are known and stationary, the optimum
filter coefficients can be determined using a set of Normal Equations.
•    If we don’t know the statistics exactly (or if they are time-varying), we
need an adaptive filter
d(n)

+
y(n)
x(n)               f(n)                        Σ         e(n)
-

Digital Signal Processing in 2700 seconds
Applications of Optimal Filters

•    System identification - generate linear model of unknown system

d[n]
g[n]=?

x[n]
+
y[n]
f[n]                    Σ           e[n]
-

•    Linear prediction - estimate the future value of a signal
x[n]

+
x[n]
x[n-n o]      f[n]                          Σ      e[n]
-

Digital Signal Processing in 2700 seconds
Adaptive Filter Application Examples
•    Adaptive echo cancellation

+ Σ              delay

-
Filter                                             Hybrid
Filter

-
delay                  Σ
+

•    Adaptive line enhancement (detect small periodic signals buried in noise)
+       Σ
e[n]
x[n]
-

z-m                                      y[n]
Filter

Digital Signal Processing in 2700 seconds
DSP Processors

•    DSP processors are optimized for Multiply/Accumulate (MAC) operations.
•    Multiple data/program busses inside the chip allow simultaneous access to
program and data memory (Harvard Architecture).
•    Modern DSP chips can implement up to 8 instructions in a single clock cycle.

Processor     Processor                 Program    Data
Unit          Unit                   Memory    Memory

Program Bus
Data Bus

Digital Signal Processing in 2700 seconds
DSP Processor Performance
•    Digital Signal Processor chips are amazingly fast!
– TI C67 (\$200): 32-bit floating-point, 1GFLOP
– ADI SHARC (\$60): 32-bit floating-point, 150MFLOP
1024 Point Complex Radix 2 FFT with bit reversal
Speed
Processor                                   (usec)
ADI TigerSHARC @ 150MHz                          69
TI C67 @ 167MHz                                124
TI C40 @ 50 MHz                               1435
Power PC 604e @ 333 MHz                        230
Intel Pentium @ 200 MHz                        750
Vax 8600                                     21700
Other DSP operations (Based on TI C67 @ 167 MHz)
Speed
Operation                                (usec)
Matrix-Vector Multiply 38x160 * 160x1       41.2
Autocorrelation, 18 x 8                    0.606

Digital Signal Processing in 2700 seconds
DSP Performance
•    BUT!
– To achieve the benchmark performance:
• Algorithm must run in a tight loop
• Processor pipeline must be kept full
• Code for Algorithm must fit in on chip cache
• Data arrays must be within on chip cache
• Parallel execution must be maximized
– Scatter/Gather operations will suffer performance degradation.

Digital Signal Processing in 2700 seconds
Effort Required to Achieve A/D Performance
6

8

10
Effective Number of Bits

Relatively Easy
s)
12                                                               pec
eS
nc
14
orma
Perf
16                                              DC
Specialized                          (A
18       Knowledge
lt
fficu
20                           Di
Difficult to
22
Impossible
24

26
1      10      100      1K      10K 100K           1M      10M 100M        1G
Sample Rate (Hz)
Ref: “Practical Limits of Analog-to-Digital Conversion” (Jerry Horn)

Digital Signal Processing in 2700 seconds
Conclusions

•    Applications for digital signal processing are exploding, largely fueled by the
availability of inexpensive high performance processors.

•    There are certain applications where digital is clearly better than analog.

•    The accelerator community is starting to tap the capabilities of DSP
technology, but this is just the beginning...

Digital Signal Processing in 2700 seconds

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