This document presents an algorithm called Fractional Fourier Transform (FXT) to remove spectral leakage caused by non-coherent sampling of sinewaves. The algorithm works by "twisting" the time/frequency space to accommodate fractional periods. It was shown through simulations and ADC testing to automatically correct for frequency drift, maintain spectral resolution, and conserve SNR. The FXT algorithm allows using non-coherent oscillators for testing applications like ADC or waveform recorders.

1. Algorithm to remove spectral leakage, close-
in noise and its’ application to converter test
Dr. Fang Xu, Teradyne, Inc.
Boston, MA 02466 U.S.A.
Fang.xu@teradyne.com
st techniques to face new challenges
Developing n ew te
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2. Periodicity as Base of DFT
Uniformly sampled in first domain ⇔ Periodic in other domain
Periodic in first domain ⇔ Uniformly sampled in other domain
Uniformly sampled and periodic in first domain
⇔
Uniformly sampled and periodic in other domain
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3. Problem Statement
• Discrete Fourier Transform is based on periodic signal and assumed
that the signal is repetitive outside the interval the DFT is performed
• If during that interval, the DFT is applied to a sine-wave with
fractional period, huge artifacts around that tone can be observed.
This is called leakage
FFT of 8192.5 periods sinewave
FFT of 8192 periods sinewave
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4. DFT Applied to Periodic Waveform
Oscillation has integer number of periods
1 Discrete Fourier Transform time interval
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5. Root Cause of Spectrum Leakage
Fraction period causes signal
discontinuity, which in turn
causes spectrum leakage
Oscillation has fraction number of periods
1 Discrete Fourier Transform time interval
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6. Coherent Sampling
• Consider a single tone sinewave
~(t ) = A cos(2πft + Φ ) or ~(t ) = A e j ( 2πft +Φ ) + e − j ( 2πft +Φ )
x x
2
1
Uniformly sampled at T=
Fs
M N is total number of samples
f = M is total number of periods within NT
NT
If M is integer → coherent sampling → no leakage.
Otherwise → leakage
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7. Window Functions Reduce Leakage
2π n 2πn
wHamming (n) = 0.54 − 0.46 cos( ) wHanning (n) = 0.5 − 0.5 cos( )
M −1 M −1
Hamming Hanning
FT
Blackman
2πn 4πn
wBlaclman (n) = 0.42 − 0.5 cos( ) + 0.08 cos( )
M −1 M −1
`Window functions artificially make-
up a periodic signal in time domain to
reduce spectrum leakage. It is
equivalent to a convolution of FT of
window function in frequency domain
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8. Window Function Effect
Reduced leakage, but peak is still larger than coherent sampling
FFT of 8192.5 periods
sinewave
FFT of 8192.5 periods
sinewave with
Hanning window
8192
periods
sinewave
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9. FXT Concept
Im
Initial signal space Re
Time/Space carrying the
t signal was initially straight
A j ( 2πft +Φ )
x (t ) = e
2
Mt
A j ( 2π NT +Φ )
or x (t ) = e
2
Number of periods M=M0+M1
Integer portion Fraction portion
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10. FXT Concept con’t
Im
Initial signal space Re
Time/Space carrying the
t signal was initially straight
Im
Re
Mt Time/Space carrying the
Twisted signal space − j 2π 1
e NT signal has been twisted to
t accommodate fractional
period
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11. FXT Algorithm
Step 1: Perform Fourier Transform
FT
0
Time domain Frequency domain
Step 2: Locate fundamental bin M 0
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12. FXT Algorithm con’t
Step 3: Perform Inverse Fourier Transform
FT-1
0
Time domain Frequency domain
M0
− j 2π i
Step 4: Multiplication by e N
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13. FXT Algorithm con’t
Step 5: Compute phase of each point
I ϕ
R
tan-1
Step 6: Compute phase difference ∆P=2πM1
M1
− j 2π i
Step 7: Multiplication by e N
Fractional period
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14. FXT Algorithm con’t
Step 8: Compute Fourier Transform
Time domain
FFT of 8192 periods and
FXT of 8192.5 periods
FT
Frequency domain
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15. Validation with ADC
Digital signal
Synthesizer 2 ADC capturing and
processing
Synthesizer 1
Deliberately changing synthesizers frequency during SNR measurement
No difference observed – Conforming to simulation
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16. Application to ADC Test
Digital signal
Oscillator ADC capturing and
processing
Synthesizer
With correction
mean = 65.42dB
σ = 0.24dB
Without correction
mean = 64.54dB
σ = 0.86dB
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17. Comparison with Other Methods
Windowed Fourier Transform
• Reduced spectral resolution
Window • May not conserve SNR
Interpolated Fourier Transform
Naturally limited to Nyquist band
FXT • Automatic oscillator drift
correction
Im Im
Re • Identical spectral resolution
Mt
Re e
− j 2π 1
NT than coherent sampling
t t • Conservation of SNR
• Ideal for sinewave
Change the way we observe the signal • Small challenge for multi-tone
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18. Conclusion
• An algorithm to remove spectral leakage for none
coherently sampled sinewave has been presented
• Both simulation and experiments on real hardware of this
algorithm provide excellent results
– Automatically correct frequency drift
– Identical spectral resolution as coherent sampling
– Conservation of Signal-to-Noise Ratio
• This algorithm allows the use of free running crystal
oscillator to test ADC or waveform recorder
• Hope to see this algorithm been applied to other
applications
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