Chapter 7 - DSP Based Testing

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Chapter 7 - DSP Based Testing
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Outline

• Trigonometric Fourier Series (FS)
• Discrete-Time Fourier Series (DTFS)
• Relationship to FS
• Working directly with samples
• Complex form
• Discrete Fourier Transform (DFT)
• Fast Fourier Transform (FFT)
• Applications
• Equivalence of Time and Frequency Domains
• Frequency Domain Filtering
• Summary

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• Advantages of DSP Based Testing
• Reduced Test Time
• DSP in this class will be limited to discrete
  (i.e. sampled) waveforms of finite length.
• Advantages of coherent DSP based testing
• reduced test time since we can create signals
  with multiple frequencies at the same time.
• Once the output response of the DUT has been
  captured using a digitizer or capture memory, DSP
  allows the separation of test tones to give
  individual gain and phase measurements.
• Also, by removing the input test tones, we can
  measure noise and distortion without running many
  separate tests.

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• Advantages of DSP Based Testing
• Separation of Signal Components
• By using coherent test tones, we are guaranteed
  that the harmonic distortion components will fall
  neatly into separate Fourier spectral bins rather
  than being smeared across many bins.
• DSP based testing also has the major advantage in
  the elimination of errors and poor repeatability.
• Advanced Signal Manipulations
• DSP allows us to manipulate digitized output
  waveforms to achieve a variety of results
• We can apply mathematical filters to remove noise
  thereby achieving better accuracy.

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• Digital Signal Processing
• DSP and Array Processing
• There is a slight difference between array
  processing and Digital Signal Processing.
• An array (or vector) is a series of numbers (i.e.
  height of students in class)
• A Digital Signal is also a series of numbers
  (i.e. voltages), yet the series is time stamped
• Thus digital signal processing is a subset of
  array processing using time-ordered samples.
• All DSP is accomplished on a special computer
  called the array processor (so much for the
  difference)

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• Digital Signal Processing
• DSP and Array Processing - cont.
• Array processing functions that are useful in
  mixed-signal testing
• averaging
• To measure the RMS of a signal we must first
  remove the DC offset - this is accomplished by
  averaging the signal and subtracting the result
  from the original
• Many functions like averaging are built into the
  ATE tester code set to allow easy use.
• Built in functions are set up to maximally
  utilize the available computational resources to
  reduce test time.

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• Digital Signal Processing
• DSP and Array Processing - cont.
• Other built in functions include
• vector average - average value of an array
• vector RMS - root mean square of the array values
• max/min - maximum and minimum values in an array
• vector add - add two arrays
• add scalar to vector - add constant to each array
  value
• subtract scalar from vector - subtract constant
  from each array value
• vector multiply - multiply two arrays
• multiply vector by scalar - multiply each array
  element by a constant
• divide vector by scalar - divide each array
  element by a constant

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• Discrete Fourier Analysis
• Fourier Transform
• Jean Baptiste Joseph Fourier
• French mathematician that found that any periodic
  waveform can be described as the sum of a series
  of sine and cosine waves at various frequencies
  plus a DC offset.
• Developed for the study of heat transfer in solid
  bodies
• A sequence is assumed to be periodic with a
  period T such that x(t) x(t-T) for all values
  of t from minus infinity to plus infinity.
• x(t) a0a1cos(w0t)b1sin(w0t)a2cos(2 w0t)
  to infinity

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• Discrete Fourier Analysis
• Discrete Fourier Transform
• Mathematical operation that allows us to split a
  composite signal into its individual frequency
  components.
• A DFT operation is equivalent to a series of very
  narrow band pass filters followed by
  peak-responding voltmeters. The filters are not
  only frequency selective but also phase selective
  to determine the sine and cosine contributions
  individually.
• x(n) a0a1cos(2?n/N)b1sin(2?n/N)a2cos(2?n/N
  ) a(N/2)cos(2?(N/2)n/N) b(N/2)sin
  (2?(N/2)n/N)

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The DFT corresponds to a bank of filters and
meters
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• Discrete Fourier Analysis
• Discrete Fourier Transform - cont.
• Digitizing spectrum analyzers and mixed-signal
  testers accomplish the filter and peak
  measurements using the DFT. The DFT uses a
  frequency sensitive correlation calculation for
  each value of a and b.
• Functions that have zero correlation are called
  orthogonal
• Superposition and orthogonality of coherent sine
  and cosine components allows us to extract the
  value of all as and bs, even in the presence of
  other coherent test tones. The cosine correlation
  function is equivalent to a filter and peak
  measurement. Therefore we can measure many
  signals simultaneously, reducing test time.

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Frequency and Phase Selectivity of DFT
Correlations
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• Discrete Fourier Analysis
• Complex form of the DFT
• Most traditional DSP books use the Eulers
  transform to convert sinusoids into exponentials.
• e-j w t cos(wt) jsin(wt)

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• Discrete Fourier Analysis
• Complex form of the DFT
• Notice that the complex form of the DFT
  correlates with a negative sine wave instead of a
  positive sine wave in the sine/cosine version.
• This causes problems in the phase shift
  calculations!!!
• Some testers will give the straight imaginary
  value, while others multiple by minus one to
  compensate for the difference.
• The test engineer will need to find out whether
  the tester is reporting sine amplitudes or
  imaginary components before phase measurements
  can be made!!!

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Fourier Analysis Of Periodic SignalsTrigonometric
Form

• For any periodic signal with a finite number of
  discontinuities, the signal can be represented by
  a Fourier Series

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Computing Fourier Coefficients

• Coefficients are found from the following
  integral equations

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Fourier Series RepresentationMagnitude Phase
Form
Rectangular Form
MagnitudePhase Form
where
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Spectral Plot

Phase
f1
f3
f
0
f2
fo
2fo
3fo
4fo
5fo
0
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Fourier Series ExampleClock Signal
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Fourier Series ExampleClock Signal
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Spectrum of Clock Signal Example
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Actual Vs. FS Representation

• Increasing the number of terms in the FS
  increases the accuracy of the representation.
• Gibbs phenomenon (overshoot at discontinuity) is
  a result of the finite sum of terms.

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Outline

• Trigonometric Fourier Series (FS)
• Discrete-Time Fourier Series (DTFS)
• Relationship to FS
• Working directly with samples
• Complex form
• Discrete Fourier Transform (DFT)
• Fast Fourier Transform (FFT)
• Applications
• Equivalence of Time and Frequency Domains
• Frequency Domain Filtering
• Summary

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Discrete-Time Fourier SeriesFirst Principles
Consider sampling x(t)
But, FS1/TS, allowing us to write
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Discrete-Time Fourier SeriesCoherent Sampling

• Generally, we are interested in only those sample
  sets that are derived from a signal that
  satisfies TNTS or foFS/N

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Discrete-Time Fourier SeriesPeriodic Sample Sets

• The fact that we are using coherent sample sets,
  implies periodicity in n. However, due to the
  symmetry of the formulation, xn is also
  periodic with respect to k with period N

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Discrete-Time Fourier SeriesRe-Grouping
Formulation
Split into 2 parts
To simplify further, use trig. substitutions
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Discrete-Time Fourier SeriesRe-Grouping
Formulation
Replace infinite summations with single
parameter
DTFS
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Discrete-Time Fourier SeriesMagnitude Phase
Notation
Rectangular Form
MagnitudePhase Form
Used for spectral plot purposes
where
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DTFS ExampleClock Signal
Evaluate Infinite Summations
After 100 terms
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DTFS ExampleClock Signal
DTFS
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FS Versus DTFSClock Signal Example

• Unlike a FS that attempts to represent the
  periodic function over all time, a DTFS only
  attempts to represent the N periodic samples
• Hence, a much simpler mathematical expression.

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Working Directly With DTFS
Strategy to solve for unknown parameters -Each
sample must satisfy the DTFS for xn

• A DTFS has N unknown parameters corresponding to
  N degrees of freedom.
• A DTFS is a representation for a coherent sample
  set consisting of N samples.

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Solving N Equations In N Unknowns
1st sample (n0)
2nd sample (n1)
Nth sample (nN-1)
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Matrix Formulation Solution
Compact notation
Unknown parameters
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Method of Orthogonal Basis

• Even before Fouriers development in the 1800s ,
  the famous mathematician, Euler had developed a
  closed-form solution for finding the unknown
  coefficients of the DTFS.
• involves projections onto a set of orthogonal
  basis functions (harmonically-related sinusoids).
• his efforts were dropped in the direction of
  Fourier analysis because of the conceptual
  difficulties that occurred with the step
  discontinuities in the signal.
• The importance of this method is that it forms
  the basis of all modern methods related to
  Fourier Analysis, Wavelets, etc.

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Method of Orthogonal Basis
DTFS Coefficients

• The above formulae are found by multiplying the
  DTFS by (i) cosk(2p/N)n (ii) sink(2p/N)n,
  then summing n from 0 to N-1.

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DTFS ExampleClock Signal
10 samples
bk coefficients
ak coefficients
All other coefficients are zero.
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Spectral PlotClock Signal Example
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Complete Frequency SpectrumHarmonics from k 0,
, infinity
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Frequency Denormalization
FS 100 kHz N 10

• DTFS is expressed in normalized time and
  frequency.
• To return to proper time scale
• To return to proper frequency scale

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Complex Form of the DTFS

• Through the application of Eulers identity, we
  can convert the DTFS in trigonometric form to the
  complex form of the DTFS,

where
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Complex Form of the DTFSSeveral Examples
Example 1
Example 2
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Outline

• Trigonometric Fourier Series (FS)
• Discrete-Time Fourier Series (DTFS)
• Relationship to FS
• Working directly with samples
• Complex form
• Discrete Fourier Transform (DFT)
• Fast Fourier Transform (FFT)
• Applications
• Equivalence of Time and Frequency Domains
• Frequency Domain Filtering
• Summary

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Discrete-Time Fourier Transform

• Fourier greatest invention was the Fourier
  Transform (FT).
• provides a frequency description (known as a
  Fourier transform) of an aperiodic signal
  (transient signal)
• If yn exists for only finite time, then we can
  represent it by the following periodic function
  xn with period N (periodic extension of yn)

yn
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Discrete-Time Fourier TransformAperiodic Signal
Description
yn

• Given some aperiodic signal yn that can be
  described in terms of a periodic signal xn,
  then we can write

• As xn is a periodic function, we can write yn
  as

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Discrete-Time Fourier TransformInvestigating
Impact of N-gt?
add zeros

• As the period N is made larger, a better match is
  made between yn xn. As N-gt?, ynxn for
  all finite values of n.
• Due to limiting argument, the infinite sum eqn.
  changes into an integral eqn

• The term Y(ejw) is called the D.T. Fourier
  Transform of yn, given by

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Discrete-Time Fourier TransformExample

• Consider a set of samples from a unit-height
  rectangular pulse signal, the F.T. would be
  computed as follows

Note Spectrum is continuous.
Y(w)
5
w
0
2p/5
4p/5
-2p/5
-4p/5
?(w)
p
4p/5
2p/5
w
0
-2p/5
-4p/5
-p
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Relationship Between DTFS FT
Y(w)
5
w
0
2p/5
4p/5
-2p/5
-4p/5

• The spectral coefficients of an N-point DTFS are
  samples of the FT

• Substituting the appropriate values for Y(ejw)
  gives

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Discrete Fourier Transform (DFT)

• The DTFS representation of the periodic extension
  of an aperiodic signal yn is referred to as a
  Discrete Fourier Transform (DFT) of yn.
• In essence, we are working with a DTFS, just
  attaching different meaning to the underlying
  result.
• The coefficients X(0), X(1), , X(N-1) are
  referred to as the DFT of yn.

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Outline

• Trigonometric Fourier Series (FS)
• Discrete-Time Fourier Series (DTFS)
• Relationship to FS
• Working directly with samples
• Complex form
• Discrete Fourier Transform (DFT)
• Fast Fourier Transform (FFT)
• Applications
• Equivalence of Time and Frequency Domains
• Frequency Domain Filtering
• Summary

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• Fast Fourier Transform (FFT)
• In the early 1960s James Tukey invented a new
  algorithm for calculating the DFT in a much more
  efficient manner.
• An IBM programmer J.W. Cooley generated the
  computer code for Tukeys algorithm and the
  Cooley-Tukey Fast Fourier Transform was created.
• Uses a folding principle called a butterfly to
  reduce the number of calculations required.
• Decimation in frequency or Decimation in Time

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Fast Fourier Transform

• The Fast Fourier Transform, or FFT, is a highly
  efficient procedure for computing the DFT/DTFS.
• For N samples, the FFT requires Nlog2N complex
  additions and (N/2)log2N complex multiplication,
  whereas the DFT requires N(N-1) complex addition
  and N2 complex multiplications.
• With N512, the FFT has a 50 to 1 advantage over
  the DFT.
• N is selected as a power-of-two (i.e., 2n), but
  other algorithms exist that can work other
  factors.

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Interpreting the FFT Output

• Most software packages, including Matlab,
  implements the following FFT algorithm
• To determine the spectral coefficients of the
  corresponding DTFS, we must perform the scaling
  operation on the samples of the Fourier
  Transform

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Interpreting the FFT Output

• DTFS in complex form
• To convert DTFS back into rectangular form, we
  use

where
and
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Interpreting the FFT Output

• To convert DTFS into magnitude and phase form, we
  use
• For RMS Value

where
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Interpreting the FFT OutputExample
Time-Domain
Frequency-Domain
Spectral Coefficients
or
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Noise Power Calculation
Spectrum (dB)
BIN

• Include only the noise power ignore the power
  contained in the signal bin and its harmonics
  (say, contained in S bins)

Correction factor
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Spectral Behavior of a Coherent versus
Non-Coherent Sinusoidal Signal
Logarithmic Scale
Coherent case (M3, f0, N64)
FS/2
Freq. Resolution
FS/2
Spectral leakage
Incoherent case (Mp, f0, N64)
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Outline

• Trigonometric Fourier Series (FS)
• Discrete-Time Fourier Series (DTFS)
• Relationship to FS
• Working directly with samples
• Complex form
• Discrete Fourier Transform (DFT)
• Fast Fourier Transform (FFT)
• Applications
• Equivalence of Time and Frequency Domains
• Frequency Domain Filtering
• Summary

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Equivalence of Time and Frequency Domain
Information
The samples of a periodic signal can be described
in matrix form as
From which the spectral coefficients are found
from
Conversely, given the spectral coefficients, the
original samples can be determined through an
inverse operation given by
This inverse operation can be computed using an
inverse FFT.
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Inverse FFT ApplicationExample
Spectral Coefficients
Time-Domain Samples
Fourier Transform Samples (N8)
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Parsevals Theorem
Complex Form for DTFS
Trigonometric Form for DTFS

• Parsevals theorem states the power of the signal
  in either the time or frequency domain is a
  constant.
• In the time-domain, both signal and noise occur
  at the same time, whereas in the
  frequency-domain, most of the noise occurs at
  frequency locations not occupied by signal.

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Applications of Inverse FFTImproving Time
Resolution of Rise/Fall Time
Noisy signal
Improved Signal (1/2 Noise)

• Knowledge of the spectral distribution of a
  signal can be exploited to improve the SNR of the
  overall measurement.
• Here the clock signal is known to consists of
  only odd harmonics, hence, by setting all even
  Bins to zero, improves SNR measurement by 3 dB.

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Applications of Inverse FFTTime-Domain
Interpolation
N Samples
Freq. Res FS/N
Freq. Res FS/(NNZ)
NNZ Samples
Add NZ zeros

• Zero-padding with Nz zeros a frequency spectrum
  consisting of N samples, followed by an IFFT,
  improves the time resolution by the factor
  (NNZ)/N.

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Outline

• Trigonometric Fourier Series (FS)
• Discrete-Time Fourier Series (DTFS)
• Relationship to FS
• Working directly with samples
• Complex form
• Discrete Fourier Transform (DFT)
• Fast Fourier Transform (FFT)
• Applications
• Equivalence of Time and Frequency Domains
• Frequency Domain Filtering
• Summary

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Applications of Inverse FFTFrequency-Domain
Filtering
x(n)
c(k)
xfilter(n)
c(k)H(ejw)
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Noise A-Weighting Filtering

• Audio measurements often call for noise
  measurement to be weighted in a manner that more
  closely approximates the frequency behavior of
  the ear.
• only the magnitude of the spectrum is of interest.

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Summary

• Coherent DSP-based testing allows AC measurements
  to be performed in near-optimum test time.
• DSP techniques involving FS, DTFS, DFT and FFTs
  were described.
• DSP-based test techniques enable test techniques
  not available with bench-top equipment, i.e.,
• Frequency-domain filtering
• Time-domain interpolation
• Noise-reduction