Digital Signal Processing _in 2700 seconds_

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					                            Digital Signal Processing
                                (in 2700 seconds)

                            John Carwardine and Frank Lenkszus
                                  Advanced Photon Source

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

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

Digital Signal Processing in 2700 seconds
                                            “Black Box” View


                                               Analog                      Analog
                                                         ADC   DSP   DAC
                                                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


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

       •    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

                                                           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.

                                                       slow orbit
                  135KHz                      point


                                               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

      •    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


              Attenuation (dB)



                                 -25      16-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
        Attenuation (dB)



                            -30                                             FIR Filter Coefficients
                            -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.
                                    1       passband

                                        0              Fc

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


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.

                     Signal Input

                                                        Fs > 6MHz
                     Local oscillator

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



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

Digital Signal Processing in 2700 seconds
                                       Analog I/Q Detector

              Signal Input
                                                                           A/D        I

          Local oscillator                  0o
                                                                           A/D        Q

      •    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.


                                            4.9MHz                        Digital
                                              I.F.             sin(wn)   lowpass
             Signal Input

                                                     19.6MHz                          Q
             Local oscillator
                                                               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]

                                                                              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

                           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


                                               f[n]                    Σ           e[n]

         •    Linear prediction - estimate the future value of a signal

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

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

                                        + Σ              delay

                        Hybrid                                              Adaptive
                                         Filter                                             Hybrid

                                                         delay                  Σ

        •    Adaptive line enhancement (detect small periodic signals buried in noise)
                                                                    +       Σ

                                                   z-m                                      y[n]

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.

                                            Program Address Bus
                                             Data Address Bus

                        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
                          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)
                          Operation                                (usec)
                          8 Cascaded Biquad filters                  0.366
                          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


                  Effective Number of Bits

                                                                Relatively Easy
                                             12                                                               pec
                                             16                                              DC
                                                      Specialized                          (A
                                             18       Knowledge
                                             20                           Di
                                                                                                    Difficult to

                                                  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

     •    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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Description: DSP (digital signal processor) is a unique microprocessor based digital signal processing large amounts of information in the device. Its working principle is to receive analog signals, convert the digital signal 0 or 1. The digital signal and then modify, delete, strengthen, and in other systems chip interprets the digital data back to analog format data or the physical environment. It is not only programmable, but in fact, speed up when tens of million complex instructions per second procedure, far more than general-purpose microprocessors, digital electronics is increasingly important world of computer chips. It is a powerful data processing capability and high speed, the two most remarkable features.