Spectral Leakage in the Discrete Fourier Transform

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Spectral Leakage in the Discrete Fourier Transform
Greg Adams, LMCO MS2, 4/10/07
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Spectral Leakage in the Discrete Fourier Transform
Synchronous Sampling is typically used with a
Discrete Fourier Transform when testing analog to
digital converters in the laboratory. A pure
sine wave test signal is generated at such a
frequency that the input signal goes through a
whole number of cycles during the sampling
period. If the test signal is slightly off
frequency, i.e. the input signal doesnt complete
a whole number of cycles within the DFT time
window, A distortion called spectral leakage
occurs. A small frequency error has little
effect on the main signal, but has a strong
effect on the DFT noise floor. The relationship
between frequency error, and the signal to noise
ratio due to leakage noise has been established.
This relationship can be used to determine the
frequency resolution which the sine wave
generator must have in order to generate a sine
wave at a sufficiently accurate frequency. A
simple calculator program is provided to evaluate
the equations.
LMCO NESS SS Math Physics Seminar 16 April
2003 Greg Adams, 856 722
4705 http//www.motown.lmco.com/gadams/
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FFT, On Frequency
fs80e6, N32768, signal freq 24 MHz, FFT Bin
size 2441 Hz
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FFT, .01 Hz offset
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FFT, 0.1 Hz offset
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FFT, 1 Hz offset
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FFT, 10 Hz offset
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FFT, 100 Hz offset
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FFT, 1000 Hz offset
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Time Discontinuity
One way of looking at the leakage problem is to
observe the requirement that the Fourier Series
operate on a periodic data set. If the
off-frequency sinusoid is repeated to generate a
periodic signal as shown, there is a
discontinuity in the waveform. The resulting
signal is not sinusoidal.
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The Fourier Series (1)
Where
The Fourier Series may be used to express any
periodic function of TIME as the sum of Sine
and Cosine functions of TIME. (expressed here in
complex exponential form) Note that the
coefficients Cn are derived by Correlating f(t)
with the Discrete frequency sinusoids sin(nt)
and cos(nt). The condition that f(t) be Periodic
insures that it can be represented as a sum of
Discrete sine and cosine functions. (whether
Laurent believed it or not! )
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The Fourier Series (2)
A Fourier series may be used, for example, to
show that a square wave is the sum of a sine
wave, and all of its odd harmonics.
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The Fourier Series (3)
While the independent variable may be something
other than time, and the Series may break the
function down in terms of any complete set of
Orthogonal functions, this discussion will
assume a function of time, Broken down into a
sum of circular functions (of time). Well also
be restricting ourselves to real-valued functions
of time.
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The Fourier Transform
The Fourier Transform is an extension of the
Fourier Series. Whereas the Fourier Series was
restricted to periodic functions of (t), the
Transform may be applied functions which
are aperiodic. While the Fourier SERIES resulted
in an infinite series of discrete frequencies,
the TRANSFORM, F(w) is a continuous function of
frequency, defined for all real values of
frequency.
A continuous function of frequency
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A table of Fourier Transforms
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The Discrete Fourier Transform
The Discrete Fourier Transform transforms a
Finite length series of Discrete time samples
f(k), into a Finite length, series of Discrete
frequency samples F(n).
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Finite Length
Because the DFT operates on a data set of finite
length, the Function f(t) must be multiplied by a
rectangular window function before Being
transformed. The window function is defined to
be One for p lt t lt p, and zero otherwise.
This is the first function Appearing in our
table of transforms, with T2p. Well denote
this window function as B(t) since its sometimes
called a Boxcar Window.
AKA the gate function, or the rectangular
function.
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Discrete time samples
Because the DFT must operate on a data set
consisting of discrete time samples, the Function
f(t) must also be multiplied by a Picket Fence
function, P(t), defined as
This product B(t)P(t)f(t) is the input data set
on which our DFT will operate.
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Series, Transform, DFT compared
Fourier Series Coefficients
DFT
Fourier Transform

• By comparing the defining equations, we can see
  that the DFT is proportional to the set of
  Fourier Series coefficients of B(t)P(t)f(t),
  with the substitution
• The DFT is integrated (summed) over an interval
  equivalent to 0 to 2p, while the Fourier Series
  is integrated over p to p.
• The terms of the DFT are equal to the integrand
  of the Fourier Transform of B(t)P(t)f(t),
  with the additional substitution.

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Table of Properties
From the table above, we see that the DFT has
more in common with the Fourier Series than the
Transform. The DFT and the Fourier Series both
have a finite time interval of integration, and
therefore yield discrete frequency samples. The
DFT alone uses discrete time samples, and is
therefore limited to a finite frequency interval
as well.
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Equivalence of DFT and Fourier Series
Since the Fourier Series coefficients Cn were
shown to be Proportional to the DFT frequency
coefficients Fn, the RATIO of signal to
integrated noise power will be identical whether
we use the DFT or the series. We will proceed to
quantify the ratio of Signal to Integrated
Leakage Noise in a Fourier Series, having proved
that this signal to noise ratio is the same
whether we use the series or the DFT. The
Integrated Leakage Noise is defined as the sum of
the noise powers, at all frequencies other than
the desired one, which result from the frequency
error.
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Notation
The traditional notation used for the DFT is
incompatible with the traditional notation used
for the Fourier Series. Well be using the
following harmonized notation
t time, seconds f(t) function of time, the
input function N Number of time samples used k
index of the kth time sample n Harmonic index,
e.g. the nth frequency bin. CnCoefficient of
the nth harmonic, Fourier Series Cs Coefficient
of the nth sideband, Fourier Series F(n)DFT of
f(k) fa analog signal frequency, Hz fs sample
rate, Hz m number or whole sine waves sampled S
sideband number P integrated noise power P(t)
Picket fence function
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Notation (cont.)
Fourier Transform of f(t)
Radian Frequency
Frequency Error, as a fraction of on frequency
BIN
Dirac's delta function

B(t) The Boxcar function (boxcar window)
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Fourier Series, with frequency error
Frequency Error, as a fraction of The frequency
resolution.
Sideband Number
Sinusoid with frequency error
Fourier Series expressed in terms of integer
frequency m, plus error.
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Evaluate the integral
Integer frequency now expressed as sideband
number S
Indefinite integral evaluated
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The arithmetic
for small delta,
and
We now have an approximation to Cs, the noise
amplitude in each sideband.
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Summing the Sidebands
The noise power in each sideband is proportional
to Cs2. There are infinitely many sidebands
above and below the carrier. The sidebands
which fall outside the normal frequency range
alias back into the output spectrum, so we must
sum Cs2 for all S not zero, from positive to
negative infinity. Since Cs2(-Cs)2, we
can Just sum from 1 to positive infinity, and
multiply by two.
But
The above sum is equal to the Riemann Zeta
function of 2, which Euler solved Explicitly in
1736.
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Signal to Noise Ratio
If some sidebands are to be omitted in the
calculation of SNR, which is often the case, we
may use the expression
For sidebands 1 through d removed
In same units as
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Testing the theory
The Leakage Noise was estimated using the formula
from the previous page, for a signal sampled at
80MSPS, with frequency offset errors ranging from
0.01 to 1000 Hz. The FFT frequency resolution
will be 2441.40625 Hz. A sinusoid was generated
by software. The resulting sine wave was
truncated to 16 bit resolution. This signals
frequency was varied over the same range of
frequency offsets. An analog to digital
converter was set up to sample a signal at 80
MSPS. The analog input signal frequency was
adjusted over the same range of frequency
offsets. A 32756 point FFT was performed on
both the computer generated signal, and the
signal sampled by the analog to digital
converter. The resulting signal to noise ratios
are tabulated below.
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Results
FFT Bin size 2441 Hz, nearest three sidebands
removed. fs80e6, N32768, signal freq 24
MHz At .01 Hz error, the real data shows ADC
thermal noise.
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Results cont.
The computer generated data had a signal to noise
ratio of 97.08 dB. The noise on the signal
results from truncating the data to 16 bits. The
Analog to digital converter had a noise floor of
79.29 dB. The noise on this signal is thermal
noise from the analog input circuit. When the
leakage noise, as predicted in the first column,
is well below the process noise, the FFT provides
an accurate measure of process noise. When the
leakage noise is significant compared with the
process noise, the noise measured by the FFT is
the sum of the process and leakage noise. The
approximation may be used to determine what
frequency accuracy is required, or how many
sidebands must be discarded, to measure a given
signal to noise ratio within a given error bound.