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OutlineOutlineOutline

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

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.

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.

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)

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.

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

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) = a0+a1*cos(w0t)+b1*sin(w0t)+a2*cos(2 w0t)… … + to infinity

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) = a0+a1*cos(2n/N)+b1*sin(2n/N)+a2*cos(2n/N)… … +a(N/2)*cos(2(N/2)n/N) + b(N/2)*sin (2(N/2)n/N)

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 a’s and b’s, 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.

Discrete Fourier Analysis

- Complex form of the DFT
- Most traditional DSP books use the Euler’s transform to convert sinusoids into exponentials.
- e-j w t = cos(wt) – j*sin(wt)

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!!!

k f

DC

f

2 f

3 f

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:

Computing Fourier Coefficients

- Coefficients are found from the following integral equations:

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.

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

Discrete-Time Fourier SeriesFirst Principles

Consider sampling x(t):

But, FS=1/TS, allowing us to write

TS

Discrete-Time Fourier SeriesCoherent Sampling- Generally, we are interested in only those sample sets that are derived from a signal that satisfies T=NTSor fo=FS/N:

TS

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, x[n] is also periodic with respect to k with period N:

Discrete-Time Fourier SeriesRe-Grouping Formulation

Split into 2 parts:

To simplify further, use

trig. substitutions:

Discrete-Time Fourier SeriesRe-Grouping Formulation

Replace infinite summations with single parameter:

DTFS:

Discrete-Time Fourier SeriesMagnitude & Phase Notation

Rectangular Form:

Magnitude&Phase Form:

Used for spectral plot purposes

where

interpolation

samples

clock signal

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.

TS

Working Directly With DTFSStrategy to solve for

unknown parameters:

-Each sample must satisfy the DTFS for x[n]

- 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.

Method of Orthogonal Basis

- Even before Fourier’s development in the 1800’s , 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.

Method of Orthogonal Basis

DTFS Coefficients:

- The above formulae are found by multiplying the DTFS by (i) cos[k(2p/N)n] (ii) sin[k(2p/N)n], then summing n from 0 to N-1.

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:

Complex Form of the DTFS

- Through the application of Euler’s identity, we can convert the DTFS in trigonometric form to the complex form of the DTFS,

where

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

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 y[n] exists for only finite time, then we can represent it by the following periodic function x[n] with period N (periodic extension of y[n]):

y[n]

Discrete-Time Fourier TransformAperiodic Signal Description

y[n]

- Given some aperiodic signal y[n] that can be described in terms of a periodic signal x[n], then we can write

- As x[n] is a periodic function, we can write y[n] as

Discrete-Time Fourier TransformInvestigating Impact of N->

add zeros

- As the period N is made larger, a better match is made between y[n] & x[n]. As N->, y[n]=x[n] 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 y[n], given by

1

F.T.

0

n

0

4

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

-2p/5

0

2p/5

4p/5

-4p/5

(w)

p

4p/5

2p/5

w

0

-2p/5

-4p/5

-p

Relationship Between DTFS & FT

|Y(w)|

5

w

-2p/5

0

2p/5

4p/5

-4p/5

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

- Substituting the appropriate values for Y(ejw) gives

Discrete Fourier Transform (DFT)

- The DTFS representation of the periodic extension of an aperiodic signal y[n] is referred to as a Discrete Fourier Transform (DFT) of y[n].
- 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 y[n].

- 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

Fast Fourier Transform (FFT)

- In the early 1960’s 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 Tukey’s 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

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 N=512, 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.

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:

Interpreting the FFT Output

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

where

and

Interpreting the FFT Output

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

where

-10

signal

-20

-30

harmonics

-40

-50

-60

-70

-80

-90

noise floor

-100

0

50

100

150

Noise Power Calculation- Include only the noise power; ignore the power contained in the signal bin and its harmonics (say, contained in S bins):

Spectrum

(dB)

BIN

Correction factor

Spectral Behavior of a Coherent versus Non-Coherent Sinusoidal Signal

Logarithmic Scale

Coherent case

(M=3, f=0,

N=64)

FS/2

Freq. Resolution

FS/2

Spectral leakage

Incoherent case

(M=p, f=0,

N=64)

- 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

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.

Inverse FFT ApplicationExample

Spectral Coefficients:

Time-Domain Samples:

Fourier Transform Samples (N=8):

Parseval’s Theorem

Complex Form for DTFS

Trigonometric Form for DTFS

- Parseval’s 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.

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.

Applications of Inverse FFTTime-Domain Interpolation

N Samples

Freq. Res

=FS/N

Freq. Res

=FS/(N+NZ)

N+NZ Samples

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

Add NZ zeros

- 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

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.

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

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