Download - FFT - Instrumentacion
-
7/29/2019 FFT - Instrumentacion
1/13
32 IEEE Instrumentation & Measurement Magazine December 2007
An Introductionto FFT and Time
Domain Windows
Part 11 in a series of tutorials ininstrumentation and measurement
Sergio Rapuano and Fred J. Harris
1094-6969/07/$25.002007IEEE
Astudent in a digital signal processing class
once asked an insightul set o questions.
They went something like this: Proessor,
can you explain why we spend so much o our time
describing signals in the requency domain? We rst ex-amined circuits in terms o their sinusoidal steady-state
response. We then went on to study Fouri-
er series and Fourier transorms, and
we are now studying sampled
data Fourier transorms, and
the Discrete Fourier trans-
orm, all based on sinusoids
and sampled data sinu-
soids. Why not use other
basis unctions? What is so
special about sinusoids?
Great questions! We wish
we had thought to ask them!
The answers are simple.
Many o the dynamic systems we
analyze, synthesize, design, develop,
and operate can be approximated by linear
time invariant systems modeled by linear, constant-
coecient, dierential equations. So now the question
is, What does that mean? It means this: i we di-
erentiate a sinusoid, it is still a sinusoid. I we orm a
weighted sum o derivatives o a sinusoid, the sum is
still a sinusoid. The sinusoid never stops being a sinu-
soid. No other wave shape can make that claim! The
sinusoids preserve their identity in a linear system; the
system can change the sinusoids amplitude and phase
but it cannot change its basic structure. Sinusoids are
eigenunctions o linear, constant-coeicient, dier-ential equations. As such, the sinusoids can be used to
analyze and characterize the linear system.
The collection o amplitude and phase
changes experienced by sinusoids
o dierent requencies passing
through the system compactly
describes the system. We call
this description its requency
response. The requency re-
sponse is intimately tied to
the systems transer unction
and its dierential equation.
The Fourier transorm is also
used to describe signals in the time
or spatial domains. We limit our dis-
cussion here to time domain descriptions.
Signals o interest are decomposed into a set o
complete orthonormal basis unctions, the real sines and
cosines, or, equivalently, the set o complex exponentials
[3]. The classic Fourier transorm is the mechanism that
perorms this decomposition, leading to a requency
domain description o the signal. There are several ad-
vantages to analyzing signals in the requency domain.
Sensor Input Processing Output
Display,actuators,signals,control
Authorized licensed use limited to: Centro de Investigacion Tec de las Fuerzas Armadas. Downloaded on November 2, 2009 at 11:03 from IEEE Xplore. Restrictions apply.
-
7/29/2019 FFT - Instrumentacion
2/13
December 2007 IEEE Instrumentation & Measurement Magazine 33
Probably the most evi-
dent is the relevant in-
crease in dynamic range
in comparison with time
domain methods. Frequency
domain analysis can resolve:
measure and detect the ampli-
tude and phase o overlappedsignals the amplitudes o which dier
by orders o magnitude. This powerul
characteristic makes requency domain
measurements important in many ields,
including telecommunication, instrumenta-
tion, radar, sonar, consumer entertainment, and
other systems requiring signal analysis, signal
detection, modulation, and demodulation.
The Fourier theory had originally been ormu-
lated or continuous time and amplitude signals, i.e.,
analog signals and systems. With the advent o the digi-
tal computer, in particular o the microprocessor, spectrumanalysis could be perormed in the digital domain along
with many o the other signal processing tasks perormed
by digital techniques. The sampling theorem was well es-
tablished prior to the arrival o the digital computer, and the
missing components required to perorm signal processing
on sampled data signals were the transducers to pass sig-
nals back and orth between the continuous domain and the
sampled data domain. The need or high-perormance, low-
cost analog-to-digital converters (ADCs) and their duals,
the digital-to-analog converters, was recognized early on,
and their development closely paralleled that o the micro-
processor. It is interesting to note that the rst applicationso sampled data signal processing occurred in the sampled
data control area because the required sample rates were
low, as were the bandwidths o the mechanical systems being
controlled. The next application was that o processing audio
signals, also with a relatively low bandwidth matched to
the then-improved perormance capabilities o early ADCs.
This evolution is seen in the publication history, with the rst
venue or signal processing articles being the IEEE Journal on
Audio and Electroacoustics (1965), which morphed intoAudio,
Speech, and Signal Processing (1984), on its way to becoming
the Transactions on Signal Processing (1991).
The mathematical tools were ready and waiting as the
enabling technology and applications came together in the
early 1960s. When digital data (the sampled and quantized
representation o an analog signal) were delivered by ADCs to
the digital domain, they ound a rich body o signal processing
options ready to manipulate and extract signal parameters.
The Z-transorm had already been developed as the sampled
data counterpart to the Laplace transorm, and the discrete
Fourier transorm (DFT) had been developed as a counterpart
to the Fourier transorm (FT). One modication to the DFT was
required to perorm machine computation o the sampled data
spectrum. The DFT is dened as a sum over a two-sided inter-
val rom to +. In practice we oten change the lower limit o
the sum to 0 to acknowl-
edge causality and to
obtain the one-sided DFT
as a sum rom 0 to +. The
practical consideration is that
we can never perorm an ini-
nite sum in a computer. We have to
limit the range o the sum to a niteinterval, say, 0 to N 1, a sum contain-
ing Nterms. This little detail essentially
turns the data collection process on at in-
dex 0 and then turns it o at index N 1. This
gating operation means that the sum is always
perormed over nite data collection apertures,
with boundary conditions equivalent to an abrupt
turning on and o o the collected data. Turning data
abruptly on and o has an undesired infuence on the
spectrum o the collected signal samples. We ameliorate
this eect using a multiplicative weighting term applied to
the data in the collection interval, which slowly and gentlyturns the data on and o at the boundaries. This is a common
operation in many signal processing systems acing nite ap-
ertures. In time-series signal processing we call the weighting
unction a window, in spatial processing (beam orming) we
call it a shading function, and in photolithography we call it an
apodizing function.
The DFT is implemented in digital systems by a amily o
algorithms collectively known as the ast Fourier transorm
(FFT). The FFT oers a signicant reduction in computational
workload relative to the DFT. The DFT requires on the order
oN2 complex operations (multiplies and adds), while the
FFT can be implemented with workloads between 2 Nandlog2(N) N/2 complex operations. Some inormation on FFT
history can be ound in [4]. The FFT is widely applied in digital
signal processingbased systems. [5] These include modula-
tion and demodulation applications in telecommunication
systems, with examples being Orthogonal Frequency Domain
Multiplexing (OFDM) and Asymmetric Digital Subscriber
Lines (ADSL) and measurement and instrumentation systems,
the emphasis o this paper. Processing a nite aperture obser-
vation introduces several arteacts into the spectral analysis
process. One important arteact is spectral leakage, the spill-
ing o energy centered at one requency into the surrounding
spectral regions. This eect limits our ability to reliably detect
low-level signal in the presence o nearby high-level signals.
Windows are designed and applied to suppress this arteact.
A second attribute brought to bear by the nite aperture is the
uncertainty principle. The nite aperture limits spectral re-
solvability, the ability to detect closely spaced similar-strength
tones as individual signal components [3]. What we will learn
here is that when we apply a window to the signal to control
biases due to spectral leakage, we increase the width o the
spectral main lobe, which causes secondary eects related to
spectral resolution (separating nearby signals), processing
gain (separating signals rom noise), and window overlap (sat-
isying Nyquist). We will examine and highlight these eects.
What is so special about
sinusoids?The sinusoid
never stops being a
sinusoid. No other
waveshape can
make that
claim!
S en so r I np ut Processing Output
Display,actuators,signals,control
Authorized licensed use limited to: Centro de Investigacion Tec de las Fuerzas Armadas. Downloaded on November 2, 2009 at 11:03 from IEEE Xplore. Restrictions apply.
-
7/29/2019 FFT - Instrumentacion
3/13
34 IEEE Instrumentation & Measurement Magazine December 2007
The paper is intended to give the basic elements o FFT-
based spectrum analysis, starting with the properties o the
DFT and introducing considerations related to its ast imple-
mentation. We examine the deault window, the rectangle,
and its contribution to spectral leakage, as well as designed
windows and how they can be used to obtain accurate spec-
tral estimates. Finally, we discuss and compare the main
characteristics o the windows used in spectrum and networkmeasurements along with inormation on their secondary
attributes.
The Discrete Fourier TransormThe mathematic tool or analyzing signals and systems
in the requency domain is the FT. When applied to a real
or complex valued analog signal xa(t), it produces its as-
sociated requency domain representation, Xa(f), called its
spectrum. The basic idea behind spectrum analysis is that,
subject to easy-to-satisy restrictions, a signal can be ormed
as a weighted sum o complex exponential unctions (called
spectral components). The weighting terms at each requencyare the complex amplitude and phase. Spectral analysis is the
process by which we estimate the amplitude and phase o the
components located at each requency. For this reason, Xa(f)
is a complex signal, the magnitude o which represents the
sinusoids amplitudes versus requency, which we denote
the magnitude spectrum oxa(t), and whose phase represents
the sinusoid phases versus requency, which we denote the
phase spectrum oxa(t).
An aperiodic signal has requency domain representation
dened by the FT integral shown in Equation 1, as ollows:
(1)
with an inverse transorm integral as shown in Equation 2:
(2)
When xa(t)is periodic in T0s, the spectral components are
harmonics located at multiples o the undamental requency,
1/T0Hz. The signal xa(t) can be represented, in the mean square
sense, with a countable sum o sine waves by the Fourier Series
(FS) [1] integral shown in Equation 3. The resulting magnitude
and phase spectra reside only at the integer multiples o the
requency 1/T0, or k/T0.
(3)
with an inverse transorm sum as shown in Equation 4:
(4)
Note that within a scale actor, the FS (Equation 3) o a
periodic signal is a sampled FT (Equation 1) o a single cycle
o the periodic signal, as is shown in Equation 5. What we
realize here is that sampling in the requency domain induces
periodicity in the time domain. The converse, o course, is also
true: sampling in the time domain induces periodicity in the
requency domain and results in the sampled data Fourier
transorm (SDFT):
(5)
The FT and the FS are applicable to analog signals. That
means that FT and FS describe continuous time signals and
have continuous and discrete spectra, respectively, with
arbitrary amplitude and phase proiles. All mathematics
associated with the Fourier theory is applicable to digital sys-
tems. The SDFT is dened or sequences o arbitrary length
and is the counterpart o the inverse FS. By the application o
appropriate windows, the SDFT can process nite length se-
quences. The SDFT is shown in Equation 6. Here the variable is digital requency with units o radians/sample, bounded
by the interval p
-
7/29/2019 FFT - Instrumentacion
4/13
December 2007 IEEE Instrumentation & Measurement Magazine 35
(9)
The inverse DFT, which is the nite sum replacement o
Equation 8, is shown in Equation 10.
(10)
An important observation is that X(k)is a periodic set o
requency samples due to the original time domain sampling
process. The positive requencies are located at indices 0 k
N/2, and since index Nis congruent to index 0, the negative
requencies 1 k N/2 are located at N 1 kN/2.Simi-
larly, x(n) has become a periodic set o time domain samples,
periodic in Nsamples, due to the sampling o the spectral
unction.Note that quantization o sampled data sequence has no
bearing on the properties o the SDFT. The quantization noise
can be attributed to source coding noise and is treated as addi-
tive noise, which is seen as one o the noise terms in the spectra
when the spectra is computed by nite precision arithmetic
processors.
Summarizing this section, we can say that
Sampling an analog signal at requency FS induces peri-
odicity o its spectrum with period FS or with normalized
period 2p.
The DFT orms equally spaced samples o the periodic
sampled data spectrum.Sampling the periodic spectrum o the input sequence
at multiples oFS/Ninduces periodicity o the input se-
quence with period N.
The samples in the requency domain are equally spaced
at multiples oFS/N; thereore, spectral values dier-
ent rom kFS/Nare not available rom the N-length
transorm. The requency interval FS/Ndenes the
transormfrequency resolution, and the requencies
kFS/Nare denotedfrequency bins.
Additional requency resolution can be
obtained or a sequence o length N by
perorming a 2Nor 4N-point transorm
on the zero-extended versions o the
sequence.
Applications o the DFTBy using the DFT and its properties, it
is possible to do the ollowing:
Compute the spectrum o a
sampled data signal.
Perorm circular con-
volution o two sam-
pled signals as the
N-point IDFT o
the spectral product o their N-point DFTs. One o the
sequences may have to be zero extended to length N.
Perorm linear convolution o two sampled signals as the
2N-point IDFT o the spectral product o their 2N-point
DFTs. Both N-point sequences must be zero extended to
length 2N.
Determine the requency response o a sampled data
system as the ratio o the N-point DFTs o the windowedoutput and input series.
Synthesize the time-domain sampled waveorm rom its
requency domain representation.
The main advantage o the DFT over the FT is that it permits
machine computation o spectra.
The DFT is a vector process, converting input time vectors
o length Ninto output requency vectors o length N. One
matter o concern in applying the DFT is the high computa-
tional burden oN2 complex operations to convert the input
vector to the output vector. A second matter o concern is that
the spectral sampling inherent in the DFT describes the peri-
odic extension o the input signal, which may produce spectralarteacts as a result o the articial boundaries.
The rst concern has been put to rest by the development o
ecient FFT algorithms, which require signicantly reduced
computational resources to perorm the transorm. The second
concern is handily addressed by the use o windows to sup-
press the boundary conditions, which in turns suppresses the
spectral leakage arteacts.
Efcient Calculation o DFT,the FFT AlgorithmThe DFT is seen to be a set o projections o the input time
series onto the basis vectors WnkN, where WN is the Nth root ounity, . The projections are perormed as N-point inner
products, with each projection requiring Ncomplex multiply
and add operations. Thus, N2 complex multiplications are re-
quired or the direct computation o the DFT. FFT algorithms
exist or any composite length NDFT. By composite we mean
Nis actorable into a product o actors: i.e., N= N1 N2. For
instance, consider the actors oN= 800 with one possible
actoring o 32 and 25.
Many FFT algorithms operate by the divide-and-
conquer method. In this process, the transorm is
partitioned into a sequence o reduced-length
transorms that are collectively perormed with
reduced workload. For instance, a data vector
o length N= N1 N2 is mapped to a two-di-
mensional intermediate array o dimen-
sion N1-rows by N2-columns. We per-
orm a two-dimensional transorm
by N1 transorms over the row vec-
tors o length N2, which requires
N1N22 operations ollowed by
N2transorms over the col-
umn vectors o length N1,
which requires N2
N1
2
operations. Depend-
Themain
advantage
o the DFT
over the FT is that
it permits machine
computation o spectra.
S en so r I np ut Processing Output
Display,actuators,signals,control
Authorized licensed use limited to: Centro de Investigacion Tec de las Fuerzas Armadas. Downloaded on November 2, 2009 at 11:03 from IEEE Xplore. Restrictions apply.
-
7/29/2019 FFT - Instrumentacion
5/13
36 IEEE Instrumentation & Measurement Magazine December 2007
ing on the details o the one-dimensional to two-dimensional
mapping, there may be a set o phase rotations, called twiddle
actors, applied to the output o the rst set o transorms prior
to application o the second set. The total workload or the
two-dimensional transorm is seen to be N1N22 + N2N1
2 = (N1N2)
(N1 + N2), which, simply stated, is the sum o the actors times
the product o the actors. The workload or the original, un-
actored DFT is N2 = (N1N2)(N1N2). The ratio o the workloads
is seen to be the sum o the actors divided by the product o
the actors, R = [(N1N2)(N1 + N2)/(N1N2)(N1N2)] or R = (N1 +
N2)/(N1N2). With the actors or N= 1,000 o 32 and 25, this
ratio is seen to be (32 + 25)/(32 25) = 0.071. Here we see that
the actored orm requires only 7.1% o the workload o the
unactored orm, a savings o nearly 93%. The actoring pro-
cess is nested, with the 25-point transorms perormed as two-
dimensional transorms over 5-by-5 arrays and the 32-point
transorms perormed as 2-D transorms over 4 8 arrays.
A common-length FFT is known as the radix-2 DFT, which
perorms successive halving o the length NDFT, with Nbeing
a power o 2, such as 210 = 1,024. The partition described above
starts with a two-dimensional array o length N/2 by 2, which
results in a savings ratio o (512 + 2)/(512 2) = 0.502, a sav-
ings o nearly 50%. Each successive iterated partition results
in successive 50% workload savings. The radix-2 transorm
requires twiddle actors between successive partitions. The
iterative partitioning operation can be repeated until the proc-
ess perorms 512 two-point transorms, which are combined to
obtain 256 our-point transorms, which in turn are combined
to obtain 128 eight-point transorms, continuing until two
512-point transorms are combined to orm the 1,024-point
transorm [1]. Each successive combining operation requires
N/2 complex multiplications. The number o combining steps
is equal to the number o times the array length can be parti-
tioned 1 to 2 (or halved), which is log2(N). Consequently, the
total number o complex multiplications required to calculate
a radix-2 FFT is N/2 log2(N).
To give an idea o the computational eciency o the FFT
algorithm over the direct DFT calculation, consider the DFT
o length 1,024. The savings ratio is [N/2 log2(N)]/[N2] or
log2(N)/(2N) = 10/2,048 or 0.0049, a savings o 99.5%. The
actual numbers are 5,120 operations or the FFT, compared to
1,048,576 operations or the direct DFT computation.
The output o an FFT is a vector o length N, obtained by
sampling the periodic spectrum o the sampled data signal at
the requencies kFs
/N. It is common to call these spectral lo-
cationsfrequency bins. The periodic spectrum is visualized on
Fig. 1. Log magnitude plots of the FFT of complex and of real noisy sine wave at 1.7 kHz sampled at 10 kHz with spectral cuts at Fs and at Fs/2.
Authorized licensed use limited to: Centro de Investigacion Tec de las Fuerzas Armadas. Downloaded on November 2, 2009 at 11:03 from IEEE Xplore. Restrictions apply.
-
7/29/2019 FFT - Instrumentacion
6/13
December 2007 IEEE Instrumentation & Measurement Magazine 37
a circle o circumerence Fs or in normalized units o 2pf/Fs. As
we unwrap the circle to map it to a line, we traditionally cut
the circle at angle 0 so that the line extends over the interval
[0 to 2p), closed on the let, open on the right. The spectrum
obtained by this cut has requency 0 at the ar let, positive
requencies extending to the right until the mid-point, at
which point we have the hal sample rate, beyond which we
have the negative requencies. The FFT computes samples o
the spectrum at increments o 2p/N, indexing the samples
rom 0 to N 1. Sample Nis congruent to sample 0 and is, in
act, the start o the next spectral period. Thus, sample kis
the same as sample N kand the requencies with negative
index; the negative requencies are located in the second hal
o the interval. It is common in spectral displays to redene
the cut on the circle to be located at angle p to redene the
interval as (pto p). In this orm the center o the display is 0
requency, with negative requencies located to the let and
positive requencies located to the right. MATLAB uses the
command fftshift to redene the spectral cut. The result o
FFT calculation is a double-sided spectrum mapped to the
requency bins rom 0to N 1. In the redened spectral cut
the indices are mapped to N/2to +N/2 1,or in normal-
ized coordinates, (0.5 to +0.5). For complex signals the DFT
is asymmetrical and the entire span (0.5 to +0.5) is pre-
sented, and or real input signals, the DFT is Hermetian sym-
metric, so a reduced span o (0 to +0.5) is presented, since the
positive requencies, the rst N/2 + 1samples, are sucient
to represent the signal spectrum. Figure 1 presents the log
magnitude spectrum ormed by an FFT o a complex and o a
real 1.7 kHz sine wave with AWGN (Additive White Gauss-
ian Noise) sampled at 10 kHz. Here we see the spectra with
and without the redened spectral cuts. As can be seen, the
spectrum o the complex signal is asymmetric and the spec-
trum o the real signal is symmetric about both Fs/2 and 0.
Spectral LeakageAs described earlier, the DFT computes samples o the pe-
riodic spectrum X(q) associated with the N-point sampled
data sequence {x(n)}. Sampling in the time domain causes
replication (or periodic extension) o the spectrum, with rep-
Fig. 2.Top; analog 2 kHz sine wave and spectrum; middle: 3 ms rectangular window and spectrum; bottom: windowed sine wave and spectrum.
S en so r I np ut Processing Output
Display,actuators,signals,control
Authorized licensed use limited to: Centro de Investigacion Tec de las Fuerzas Armadas. Downloaded on November 2, 2009 at 11:03 from IEEE Xplore. Restrictions apply.
-
7/29/2019 FFT - Instrumentacion
7/13
38 IEEE Instrumentation & Measurement Magazine December 2007
licas positioned at multiples o1/T, the reciprocal o spacing
between time samples (T=1/Fs). Similarly, sampling in the
requency domain causes replication (or periodic extension)
o the time series, with replicas positioned at multiples oNT,
the reciprocal o the spacing between the requency samples
(Fs/N=1/NT). This property can be illustrated by computing
the N-point DFT o a sampled sequence that contains an inte-
ger number o cycles in the sequence o length N. A complexsinusoidal sequence containing an integer number o cycles in
Nsamples is dened in Equation 11.
(11)
For this sequence the periodic extension o the N-sample se-
quence is the same as the samples obtained by sampling theoriginal sinusoid, the original signal. The DFT o the sequence
has a non-zero spectral component at index k(or kcycles per
interval), indicating a single sinusoid in the time series.
The innite extent sinusoid has a FS at a single requency
k/(NT). By including distribution unctions in the FT, we rep-
resent the spectrum o a complex sine wave by a delta unction
(f k/NT). When the complex sinusoid is time limited to a
nite support o length NTseconds, its spectrum is modied.
The altered spectrum is obtained by convolving the transorm
o the innite extent version, (f k/NT), with the transorm othe truncating unction, WR(f), dened as ollows:
(12)
This unction is called rectangular window o length NT,
and its FT WR(f) is the ubiquitous unction sin(pt/NT)/(pt/
NT), commonly denoted sinc(t/NT) [1]. Figure 2 shows a
segment o the input sinusoid xa(t), the window wR(t), and
the windowed signal xa(t) WR(t), as well as their spectra
Xa(f), WR(f), and Xa(f)*WR(f), where the symbol * denotes a
convolution.The same results will be observed in the sample data
domain. The process o acquiring a inite-length record o
a signal can be modeled as
sampling and analog-to-digi-
tal conversion o an ininite
number o samples ollowed
by the multiplication o the re-
sulting innite sequence {x(n)}
by a discrete window unction
{w(n)}, the values o which are
non-zero within a speciied
span oNsamples. I, in theexample considered above, the
sampling period T=1/Fs and
the acquisition time interval Ts,
and the signal period, T0, are
chosen to satisy the relation
(13)
the FFT will have the appear-
ance shown in Figure 3. In this
example, the sampled signal is
a 2 kHz sine wave with a period
equal to 0.5 ms that is sampled
at a 20 kHz rate. The sampling
period is 50 s and covers a 3
ms time interval spanning six
cycles o the sine wave to col-
lect a total o N = 60 samples.
The FFT values will be zero
or all requency bins except
or the one corresponding to 2
kHz, because a sample is taken
at the peak o the main lobe o
Xa(f) *WR(f), while all the othersFig. 3. Top: sampled and windowed 2 kHz sine wave; middle: mid: single-sided spectrum of signal; bottom: DFTsamples of sampled sine wave.
(a)
(b)
(c)
Authorized licensed use limited to: Centro de Investigacion Tec de las Fuerzas Armadas. Downloaded on November 2, 2009 at 11:03 from IEEE Xplore. Restrictions apply.
-
7/29/2019 FFT - Instrumentacion
8/13
December 2007 IEEE Instrumentation & Measurement Magazine 39
are taken at the zeros o the same
spectrum. Note that in this case,
with the sinusoid containing
an integer number o cycles per
interval, there is no uncertainty
in estimating the sine wave re-
quency and amplitude.
Almost assuredly, we will notsee signals in which all the spec-
tral components have an integer
number o cycles per interval o
lengthN. More likely, the periods
T0are arbitrary and unknown
and there isnt an integer kthat
satises the condition (Equation
11) or a given Nand a selected
sampling period T. In these cases
the requency components cor-
responding to the FFT bins will
not correctly represent the am-plitude and spectral position o
the underlying spectral peaks
and urther will observe the side
lobe spectral terms related to the
oset spectral peaks, which spill
spectral components over all the
requency bins.
For example, let us consider
the case o the sine wave xa(t)and
suppose that T0is not known.
Let us suppose that NT = 6.5 T0,
as shown in Figure 4. The peri-odically extended version o this
sequence will have a discontinu-
ity at each Nsample not present in the original non-windowed
sequence. The FT o this sequence is the same shited sinc
unction as in the previous example, but the zero crossings no
longer correspond to the DFT sample positions. Consequently,
the DFT will now compute samples o the side lobe rather than
samples o the zero crossings. This phenomenon is called spec-
tral leakage. As with the continuous FT, the original spectrum is
the convolution oXa(f) and WR(f), with the sample positions
xed but the zero crossings translated with the center o
the sinc. Note that the sample values in the main lobe do
not coincide with the peak o the main lobe. Using this
result to estimate the signal parameters will lead to
an error in requency and amplitude. A typical peak
detection algorithm will nd a sine wave with
requency 2 kHz and magnitude 0.35, instead
o 1.1 kHz and 0.50, respectively.
The errors in measuring the signal re-
quency and amplitude are related to the
requency resolution and the main lobe
width. I the signal is composed o
many requency components with
wide variation in amplitudes,
the spectra o the high-level signals could mask or cover
the spectra o the low-level components with its side lobes.
Closely spaced spectral components may have overlapping
main lobes, which may reduce our ability to resolve them as
separate components.
Spectral leakage o window spectral side lobes is respon-
sible or (i) errors in estimating the requency and amplitude
o the requency components rom the FFT bins and (ii) a
limitation o system dynamic range as a result o the mask-
ing aect o strong components on weak components.
Spectral width o the windows spectral main lobe is
responsible or the minimum separation required
between two requency components o similar
amplitude to assure reliable detection o distinct
signal components. This separation, called
minimum resolution bandwidth, also reduces
the capability o detecting weak requency
components near strong components.
Spectral leakage cannot be avoided
when the signal components are un-
known. Proper design trades be-
tween spectral resolvability and
Fig. 4. Top: sampled and windowed 2.2-kHz sine wave; middle, single-sided spectrum of signal; bottom: DFT
samples of sampled sine wave. Note samples of spectral side lobes.
(a)
(b)
(c)
Acommon-
length FFT
is known as the
radix-2 DFT.
S en so r I np ut Processing Output
Display,actuators,signals,control
Authorized licensed use limited to: Centro de Investigacion Tec de las Fuerzas Armadas. Downloaded on November 2, 2009 at 11:03 from IEEE Xplore. Restrictions apply.
-
7/29/2019 FFT - Instrumentacion
9/13
40 IEEE Instrumentation & Measurement Magazine December 2007
detection capability are required to reduce errors in amplitude
and requency estimates.
WindowsInstead o the rectangular window, other window unctions
are generally adopted to control the limitations listed above.
All well-designed windows have the characteristic o going
gently and smoothly to near-zero values at the boundaries
o their support intervals. Consequently, the discontinuities
at the boundaries o the periodically extended sequence are
suppressed as the amplitudes and many order derivatives
are matched at the boundaries. As illustrated in Figure 5, the
window modies the input signal by smoothly and gently
bringing the signal envelope to near-zero values at the bound-
aries o the acquisition interval. The result will be a controlled
insertion o leakage.
Examining the DFT samples shown in Figure 5 we see that
the samples ar removed rom the main lobe are quite small
and will reduce the masking eects due to high side lobes. We
also see the multiple samples o the main lobe, which reduces
our ability to estimate the center requency and amplitude o
the sine wave. The known
bias in amplitude due to the
reduced-amplitude main
lobe o the Hann window
relative to the rectangle
window is easily removed.
The amplitude o each win-
dow is simply the sum othe window coeicients
and can be properly scaled
out ater the measurement.
The residual amplitude er-
ror resulting rom position
o the center lobe relative to
the DFT xed sample posi-
tions is known as scalloping
loss. This loss is smaller
than the loss obtained with
the rectangle window.
This loss can also be to-tally removed by passing
a second-order polynomial
through the three maxi-
mum-amplitude log-mag-
nitude samples in the main
lobe and by determining
the peak position and am-
plitude o the parabola.
This works amazingly well
because in log amplitude,
all spectral windows are
approximately parabolas.To reiterate, we comment
that good windows reduce
the side lobe levels, thus
improving the detectability o weak requency components,
and the errors made by estimating each component requency
and amplitude can be easily eliminated by polynomial in-
terpolation. An alternate interpolation option is obtained by
simply zero extending the length-Nwindowed time series to
length 2Nor 4Nand perorming the increased length DFT.
In Figure 6, we show examples o the time and requency re-
sponses o windows most oten used in digital spectrum analysis
[1], [6]. The choice o the window may vary or a specic applica-
tion, as each has dierent eects on the FFT output. Generally, a
window with a very narrow main lobe will have a high spectral
resolvability and a lower uncertainty in measuring the requency
o a spectral component. In most cases, a narrow main lobe implies
high side lobes causing low detectability o weak spectral compo-
nents. In addition, a narrow main lobe will cause a commensu-
rate uncertainty in the measurement o the spectral component
amplitudes as a result o high scallop loss. For example, the Hann
window is the window o choice in applications requiring high
resolvability. As a result o its narrow main lobe, the requency
resolution is maximized and the requency measurement uncer-
tainty is minimized. However, because o the higher side lobes,
Fig. 5. Effects of the Hann window. Top: windowed sine wave; middle: single-sided spectrum of windowed signal; bottom:DFT samples of windowed and sampled sine wave.
Authorized licensed use limited to: Centro de Investigacion Tec de las Fuerzas Armadas. Downloaded on November 2, 2009 at 11:03 from IEEE Xplore. Restrictions apply.
-
7/29/2019 FFT - Instrumentacion
10/13
December 2007 IEEE Instrumentation & Measurement Magazine 41
the detectability o low-level nearby spectral terms is reduced in
comparison with other windows. The uncertainty o the ampli-
tude measurement is greater than or windows designed or low
scallop loss, such as the harris Flat Top window. As a result o the
wide fat spectral response, the Flat Top has small scallop loss and
hence exhibits small amplitude errors, but the same fatness leads
to signicant requency uncertainty, which must be resolved by
one o the spectral interpolation options.
The Flat Top window, on the other hand, has a fat but very
wide main lobe and lower side lobes. The Kaiser-Bessel win-
dow, with a time-bandwidth parameter, , oers a selectable
trade between side lobe levels and main lobe width. Some
digital signal processing systems oer the option to select a
window rom a set to give the user more degrees o reedom
in the analysis. In all cases the user should be aware o the e-
ect o the windows on the measurement results. A complete
introduction to the windowing techniques as well as a review
o more window characteristics can be ound in [3].
Windows CharacterizationThere are several gures o merit used or classiying the win-
dows used in digital spectrum analysis. All parameters reer to
characteristics o the window requency response. In [3], [7], and
[8], a complete description o them is given in mathematical and
practical terms. Here some parameters are recalled and briefy
introduced in order to give some indications on how to interpret
them or nding the right trade-o or a given application.
Minimum Resolution Bandwidth
As described above, as a result o the convolution o the sig-
nal and the window spectra, the windowed signal spectrum
includes a replica o the window requency response located
at each requency component o the signal, resulting in an
Fig. 6. Time and spectral response of common window functions: rectangular, Hann, Kaiser-Bessel, and Harris Flat-Top. The width of the rectangular window
main lobe is overlapped to the other window spectra as dashed red lines to easily compare them.
Fig. 7. Overlapped components that cannot be distinguished (one peak) andthat can be distinguished (two peaks) [3].
S en so r I np ut Processing Output
Display,actuators,signals,control
Authorized licensed use limited to: Centro de Investigacion Tec de las Fuerzas Armadas. Downloaded on November 2, 2009 at 11:03 from IEEE Xplore. Restrictions apply.
-
7/29/2019 FFT - Instrumentacion
11/13
42 IEEE Instrumentation & Measurement Magazine December 2007
overlap o the main lobes corresponding to the nearest compo-
nents. As a consequence, even i the requency resolution o the
DFT is very high, it isnt possible to resolve near components
with similar amplitude whose distance is lower than the width
o the window main lobe (Figure 7). To this aim, the width o
each lobe is dened to be the requency interval correspond-ing to a 6-dB attenuation to the DC gain and is called minimum
Resolution Bandwidth (RBW) [3], [6]. As indicated in [3], the
minimum RBW goes rom 1.2 to 2.6 bins, depending on the
specic window.
Side Lobes
The detectability o weak components is mainly aected by
two characteristics o the window requency response: the
highest side lobe level and the rate o side lobe roll-o. The lev-
el o the highest side lobe is given as a ratio in dB normalized
to the main lobe level. The maximum side lobe level usually
alls inversely with the main lobe width. The side lobe roll-o
is usually given in terms o the requency response decrease in
dB per requency octave or decade. This rate is1/f(m+1),where m
is the order o the derivative o the window envelope in which
the rst discontinuity resides. For instance, a rectangle is dis-
continuous in the zero-th derivative, and its rate o spectral
decay is 1/f1 or 6 dB/octave, and a Hann window is discon-
tinuous in the second derivative and its rate o spectral decay
is 1/f3 [1], [3]. Note that the Hann window has a discontinuity
in its zero-th derivative, and its rate o spectral decay is also
1/f, but it has lower side lobes than the rectangle because it has
a smaller discontinuity. Similarly, the Kaiser-Bessel window
also has a discontinuity in its zero-th derivative, and its rate
o spectral decay is also
1/fbut has even lower side
lobes because o its even-
smaller discontinuity.
Processing Gain/Processing
Loss
The amplitude estimationo a requency component
is aected by the broad-
band noise passed by the
bandwidth o its spectral
main lobe. In this sense, the
window behaves as a lter,
gathering contributions or
its estimate over its band-
width [3]. Remember, to re-
duce side lobes we increase
main lobe width, which
permits more noise into thespectral measurement and
at the same time reduces
the amplitude o the main
lobe, which reduces the
amplitude o the desired
sine wave measurement.
The window reduces the signal-to-noise ratio (SNR) relative
to the SNR o the deault rectangle window. A measure o how
much SNR improvement obtained rom a windowed FFT can
be ound as the ratio between the SNR beore and ater the
calculation, called Processing Gain (PG):
(14)
where, So/No is the output SNR, Si/Ni is the input SNR, and w(k)
is the window sample. The PG o a rectangle window is N(sum
squared/sum o squares), the PG o a Hann window is N/1.5
and thePGor a 60-dB Kaiser Window is N/1.7. Usually thePG is
normalized to thePGo a rectangle o the same lengthN. Interest-
ingly, the reciprocal o the PG is the equivalent noise bandwidth
ENBW, the width o a spectral rectangle o unit amplitude that
passes the same noise power as the window being described.
Thus, the ENBWo a rectangle is 1/Nand o a Hann window is
1.5/N. The Hann windows equivalent lter passes hal again
as much noise as does the rectangle windows equivalent lter.
Incidentally, the reduction in SNR or PG due to use o a good
window is exactly cancelled by the variance reduction obtained
when averaging overlapped windowed transorms [8], [9].
Scalloping Loss
When the requency being analyzed by the DFT is bin-cen-
tered, the DFT sample coincides with the peak o the windows
main lobe response. When the requency is positioned oset
Fig. 8. Scalloping loss is 3.9 dB for a rectangular window and 1.2 dB for a Kaiser-Bessel window measuring a spectralpeak at position 4.5 cycles per interval, midway between bins 4 and 5.
Authorized licensed use limited to: Centro de Investigacion Tec de las Fuerzas Armadas. Downloaded on November 2, 2009 at 11:03 from IEEE Xplore. Restrictions apply.
-
7/29/2019 FFT - Instrumentacion
12/13
December 2007 IEEE Instrumentation & Measurement Magazine 43
rom the bin center, the DFT sample is oset on the main
lobe, and the measurement is less than the peak value. The
worst-case reduction in measurement value occurs when the
signal requency resides midway between two bin centers.
An example would be a sine wave with 4.5 cycles per interval
o length N, its spectral peak residing midway between bins 4
and 5. Here both adjacent bins oer a reduced-level sample o
the oset main lobe spectral response. Windows with wider
main lobe width have smaller loss as a result o the requency
oset. As seen in Figure 8, the amplitude error, called scalloping
loss, depends on the windows main lobe bandwidth. Usually
the scalloping loss is reported in dB and goes rom 0.1 to 3.92,
depending on the window [3].
Misconceptions
Common misconceptions to keep in mind when estimating the
spectrum o a signal with an FFT are the ollowing:
The transorm length Ndoes not have to be a power o 2.
FFTs come in all sizes, and any (non-prime) length FFT is
available. A 500-point transorm is as ecient as a 512 -
point transorm. Transorms with lengths that are powers
o 2 have very simple coding structure, which infuences
hardware implementations but has little bearing on sot-
ware-based spectral estimation.
A paraphrase o the Nyquist criterion tells us that the
sample rate should exceed the two-sided bandwidth o
the signal. For real baseband signals this is interpreted as
such: the sample rate must be at least twice the highest
requency component in the signal. This is a very restric-
tive interpretation and should not limit your options. The
transorm o a real signal exhibits Hermetian symmetry,
H(k) = H*(k). As such, without loss o inormation, the
FFT can be computed and displayed or positive requen-
cies only. Complex signals, on the other hand, ormed, or
instance, by a quadrature downconversion o a span o
positive requencies, does not exhibit spectral symmetry.
As such, the FFT must be computed and displayed or
both positive and negative requencies. Bear in mind that
the analog anti-aliasing lters have a transition band-
width, and not all spectral components are alias ree. It is
typical, or instance, to allow the aliased transition band
to corrupt 1020% o the spectrum so that a 1,024-point
FFT can present 400 spectral lines or real signals, or 800
lines or complex signals [2].
A number o pre- and post-processing algorithms aid
the FFT in the spectral estimation task. We discuss one
here to catch the readers interest. The spectral resolu-
tion o an FFT is initially dened by the spacing between
Fig. 9. Time domain and spectra of windows with 90-dB side lobes and main lobe width of Fs/N.
S en so r I np ut Processing Output
Display,actuators,signals,control
Authorized licensed use limited to: Centro de Investigacion Tec de las Fuerzas Armadas. Downloaded on November 2, 2009 at 11:03 from IEEE Xplore. Restrictions apply.
-
7/29/2019 FFT - Instrumentacion
13/13
44 IEEE Instrumentation & Measurement Magazine December 2007
spectral bins Fs/N. A rectangle window has a main lobe
spectral width oFs/N, which matches this spacing.
When a good window is applied to the data to suppress
the spectral side lobes, the main lobe width increases
by a actor o 4 to 4Fs/N. This increased width reduces
the spectral resolution by the same actor. A response to
this reduced resolution is to increase the data record and
window length rom Nto 4Nand to thus return the reso-lution to 4Fs/(4N).This appears to also increase the FFT
length, which we dont want to do. We keep the same-
length FFT by computing every ourth transorm point,
which matches the original spectra spacing oFs/N. The
4-to-1 spectral downsampling induces a ourold time
domain aliasing o the length 4Nwindowed signal to the
Nlength series that is processed by the N-point trans-
orm. The olding o windowed signals is perormed by
a pre-processor, prior to the FFT, known as a polyphase
partition [7][10]. The combination o olded window
and FFT is reerred to as a polyphase channelizer. Figure
9 illustrates the morphing o the windows spectral mainlobe width rom Fs/Nto 4Fs/Nand then back to 4Fs/(4N)
and6Fs/(6N).
ConclusionsThis paper includes a brie tutorial on digital spectrum analy-
sis and FFT-related issues to orm spectral estimates on digi-
tized signals. Some review o the DFT has been presented, and
some discussion on the computational advantages o the FFT
calculation has also been presented. Finally, the main consid-
erations on windowing and window characteristics have been
briefy discussed.
Reerences[1] A.V. Oppenheim and R.W. Schaer, Digital Signal Processing,
Englewood Clis, NJ: Prentice Hall, 1975.
[2] S. Rapuano, P. Daponte, E. Balestrieri, L. De Vito, S.J. Tilden, S.
Max, and J. Blair, ADC parameters and characteristics, IEEE
Instrument. Meas. Mag., vol. 8, (no. 5), pp. 4454, Dec 2005.
[3] F. J. Harris, On the use o Windows or harmonic analysis with
the Discrete Fourier Transorm, Proc. IEEE, vol. 66, (no. 1), pp.
5183, Jan 1978.
[4] M.T. Heideman, D.H. Johnson, and C.S. Burrus, Gauss and the
history o the ast ourier transorm, IEEE ASSP Magazine, Vol. 1,
(no.4), part.1, pp. 14-21, Oct. 1984.
[5] E. Brigham, Fast Fourier Transform and Its Applications, Englewood
Clis, NJ: Prentice Hall, 1988.
[6] R.A. Witte, Spectrum and Network Measurements, Englewood Clis,
NJ: Prentice Hall, 1993.
[7] F.J. Harris,Multirate Signal Processing for Communication Systems,
Englewood Clis, NJ: Prentice Hall, 2004.
[8] D. Elliot, Time domain signal processing with the DFT, Chapter
8 inHandbook of Digital Signal Processing; Engineering Applications,
Orlando, FL: Academic Press, 1987.
[9] F.J. Harris, On detecting white space spectra or spectral
scavenging in cognitive radios, in Proc. Wireless Personal
Multimedia Communications, 2007, Jaipur, India, publication in Dec.2007.
[10] F.J. Harris, Spectral analysis windowing, in Wiley
Encyclopedia of Electrical and Electronics Engineering, vol. 20,
J.G. Webster, ed., New York: John Wiley & Sons, Inc., 1999, pp.
88105.
[11] F.J. Harris, On Overlapped Fast Fourier Transorms, Int.
Telemetering Con. (ITC-78), Los Angeles, 1978, 301-306.
Sergio Rapuano (rapuano@
unisannio.it) received the M.S.
degree in electronic engineering
and the Ph.D. degree in com-puter science, telecommunica-
tions, and applied electromag-
netism rom the University o
Salerno. Since 2002, he has been
with the aculty o engineering
at the University o Sannio as
an assistant proessor in electric
and electronic measurement. Dr. Rapuano is a member o the
IEEE I&M Society TC-10 and the secretary o the TC-23 Work-
ing Group on e-tools or Education in Instrumentation and
Measurement. He is currently developing his research activi-
ties in the elds o data converters, distributed measurementsystems, and digital signal processing or measurement and
medical measurements.
Fredric J. Harris (red.harris@
sdsu.edu) is at San Diego State
University, where he teaches
courses in Digital Signal Pro-
cessing and Communication
Systems. He is a ellow o the
IEEE and author o the text
Multirate Signal Processing for
Communication Systems (Pren-
tice-Hall). He roams the world
collecting old toys and slide
rules and riding old railways.