Fundamentals of
Measurement Technology
(4)
Prof,Wang Boxiong
Digital signal processing,a field which has
its roots in 17th and 18th century mathematics,
has become an important modern tool in a
multitude of diverse fields of science and
technology,
Digital signal processing is concerned with
the representation of signals by sequences
of numbers or symbols and the processing
of these sequences,
2.3 Digital signal processing
The availability of high speed digital
computers has fostered the development of
increasingly complex and sophisticated signal
processing algorithms,and recent advances
in integrated circuit technology promise
economical implementations of very complex
digital processing systems,
The evolution of a new point of view toward
digital signal processing was further
accelerated by the disclosure in 1965 of an
efficient algorithm for computation of Fourier
transforms,the fast Fourier transform or FFT,
2.3 Digital signal processing
The fast Fourier transform algorithm
reduced the computation time of Fourier
transform by orders of magnitude,
The importance of digital signal processing
appears to be increasing with no visible sign
of saturation,
The impact of digital signal processing
techniques will undoubtedly promote
revolutionary advances in some fields of
application,
2.3 Digital signal processing
For a nonperiodic continuous time signal
x(t),its Fourier transform must be a
continuous spectrum X(f),
The continuous time signals and the
continuous spectra must be discretized first
and then truncated to get a finite length of
sequence before being processed by a
computer,This forms just the basis for the
discrete Fourier transform (DFT),
2.3.1 Discrete Fourier Transform (DFT)
dtetxfXFT ftj?2,(2.199)
dfefXtxI F T ftj?2, (2.200)
There are four cases for the Fourier transform
of an infinite-length continuous signal (Fig,
2.63),
2.3.1 Discrete Fourier Transform (DFT)
Fig,2.63 Types of Fourier transform
Fig,2.63 (a),a nonperiodic continuous
signal x(t) and its Fourier transform
spectrum X(f),The spectrum is continuous.
Fig,2.63 (b),a periodic continuous signal,
and the frequency spectrum is or discrete,
where
Δf,fundamental frequency,
2.3,1 Discrete Fourier Transform (DFT)
2
2
21:
T
T
ktfj
k dtetxTfXFT
(2.201)
k
tfj
k kefXtxI F T
2:
(2.202)
),2,1,0( kfkf k
Tf
1
Fig,2.63 (c),the Fourier transform of a
nonperiodic discrete signal,The Fourier
transform of an infinite-length discrete time
sequence is a periodic continuous spectrum,
where
Δt is the sampling period; fs is the
sampling frequency of the time sequence.
2.3.1 Discrete Fourier Transform (DFT)
n
tfj
n netxfXFT
2:
(2.203)
2
2
21,s
s
n
f
f
tfj
s
n dfefXftxI F T
(2.204)
),2,1,0( ntnt n
sf
t 1
Fig,2.63 (d),the Fourier transform of a
periodic discrete time sequence,Its
spectrum is also periodic and discrete,
The sampling period is Δt,then
2.3.1 Discrete Fourier Transform (DFT)
tNT
Conclusion:
For a periodic x(t) the spectrum X(f) is
bound to be discrete,and vice versa,
If x(t) is nonperiodic,then X(f) is
continuous,and vice versa,
The case shown in Fig,2.64 (d) where
both the time and frequency signals are
discrete and periodic provides us with the
possibility of using a computer to
implement spectrum analysis,
2.3.1 Discrete Fourier Transform (DFT)
DFT,a Fourier representation of a finite-
length sequence which itself is a sequence
rather than a continuous function,and it
corresponds to samples equally spaced in
frequency of the Fourier transform of the
signal,
where,x(n) and X(k) are periods for
and respectively,and Δt and f0 are
generalized to be unity,
2.3.1 Discrete Fourier Transform (DFT)
1,,1,0,11 2
NkWnxenxkXD FT nk
N
N
on
N
on
nkNj(2.205)
1,,1,011,1 12
NnWkX
NekXNnxI D FT
N
oK
N
oK
nk
N
nkNj(2.206)
NjN eW?2 )(? tnx?
)(? 0kfX
The true meaning of the DFT,
– It is possible to sample and truncate any
continuous time signals and make DFT of it to
get a discrete spectrum,whose envelope is the
estimate of the true spectrum of the original
continuous signal,
DFT process:
– sampling in time domain;
– truncating in time domain;
– sampling in frequency domain,
2.3.1 Discrete Fourier Transform (DFT)
2.3.1 Discrete Fourier Transform (DFT)
Fig,2.64 Graphical representation of the DFT (a)
2.3.1 Discrete Fourier Transform (DFT)
Fig,2.64 Graphical representation of the DFT (b)
2.3.1 Discrete Fourier Transform (DFT)
Fig,2.64 Graphical representation of the DFT (c)
2.3.1 Discrete Fourier Transform (DFT)
Fig,2.64 Graphical representation of the DFT (d)
1,Linearity
If
then
where a and b are constants,
2.3.2 Properties of the discrete Fourier transform
)()]([
)()]([
22
11
kXnxD F T
kXnxD F T
kbXkaXnbxnaxD F T 2121 (2.214)
2,Shift of a sequence
If
time shifting:
frequency shifting:
2.3.2 Properties of the discrete Fourier transform
)()]([ kXnxD F T?
NkmjekXmnxD F T /2][ (2.215)
mkXenxD F T Nkmj ][ /2? (2.216)
3,Symmetry property
1) If x(n) is a complex sequence,and
then
2) If x(n) is a real sequence,and
then
where XR(k) and XI(k) are the real part and
imaginary part of X(k) respectively,
2.3.2 Properties of the discrete Fourier transform
)()]([ kXnxD F T?
kXnxD FT ** (2.217)
)()]([ kXnxD F T?
kXkX
kNXkX
kNXkXkX
kNXkXkX
kNXkXkX
III
RRR
a r ga r g
*
(2.218)
3) If x(n) is an even sequence,that is,
x(n)=x(-n),then X(k) is a real sequence,
4) If x(n) is an odd sequence,that is,x(n)=-
x(-n),then X(k) is a pure imaginary
sequence,
2.3.2 Properties of the discrete Fourier transform
4,Parseval’s theorem
5,Convolution
If
then
2.3.2 Properties of the discrete Fourier transform
1 1 22 1N
on
N
ok
kXNnx (2.219)
)()]([
)()]([
kYnyD F T
kXnxD F T
kYkXnynx*
(2.220)
kYkXnynx * (2.221)
If the sampling rate is too low or the
sampling interval is too large,then the
resulted discrete time sequence may not
represent correctly the waveform of the
original signal,and frequency confusion
or aliasing will occur in frequency domain
processing,
2.3.3 Sampling theorem
2.3.3 Sampling theorem
Fig,2.65 Effect of varying the sample rate,fs,on the apparent
signal obtained by discrete sampling
2.3.3 Sampling theorem
Fig,2.65 Effect of varying the sample rate,fs,on the apparent
signal obtained by discrete sampling (continued)
The highest frequency resolved at a given
sampling frequency is determined by the
Nyquist frequency,and
To prevent these problems,the sampling
frequency should always be chosen to be
more than twice a signal’s highest frequency,
Eq,(2.223) is the sampling theorem,
2.3.3 Sampling theorem
2
s
Nyq
ff? (2.222)
ma x2 ff s?
(2.223)
2.3.3 Sampling theorem
Fig,2.66 Conditions for aliasing
(a) With frequency aliasing,when fs<2fmax
2.3.3 Sampling theorem
Fig,2.66 Conditions for aliasing
(b) Without frequency aliasing,when fs>2fmax
Windowing,to suppress or reduce leakage,
special window functions of better properties
are selected.
2.3.4 Leakage and windowing
Fig,2.67 Leakage caused by windowing for a sinusoidal function
The characteristic of a window
function:
1) 3dB-bandwidth B,the bandwidth for
which the generalized amplitude of the
main lobe remains within of its
maximum value,A generalized
amplitude of W(f):
2) Peak amplitude of side-lobe A (dB),the
ratio of the peak amplitude of side-lobe,
Asmax,to the main lobe amplitude,Am,
that is,
2.3.4 Leakage and windowing
)dB( 321?
0/lg20 WfWfW?
mm a xs AAlg20
3) Side-lobe attenuation (minimum stop-
band attenuation) D (dB/decade),the ratio
of the peak amplitude of side-lobe to the
amplitude of the side-lobe at a decade
frequency,
2.3.4 Leakage and windowing
Fig,2.68 Definition of parameters of a window function spectrum
Note:
An ideal window function must have
minimum B and minimum A as well
as maximum D,
2.3.4 Leakage and windowing
2.3.4 Leakage and windowing
Fig,2.69 Several commonly-used windows
2.3.4 Leakage and windowing
Fig,2.70 Fourier transform of windows of Fig,2.69
Picket fence effect,If the frequency fi of
a harmonic component in a signal is equal
to k/T,that is,if it coincides with the
sampling point of the output,then this
spectral line will be accurately displayed,
On the contrary,if fi does not coincide with
the sampling point,the spectral line for fi
cannot be displayed and this leads to an
error with the spectrum.
2.3.5 Picket fence effect
Frequency resolution Δf,the spacing
between two neighboring spectral lines.
When the length T of a time signal (the
window width T=NTs) and the sampling
frequency fs are fixed,so is the frequency
resolution Δf,
2.3.5 Picket fence effect
TN
ff s 1 (2.231)
2.3.5 Picket fence effect
Fig,2.72 The DFT of a periodic signal with full-period truncation
2.3.5 Picket fence effect
Fig,2.72 The DFT of a periodic signal with full-period truncation (continued)
2.3.5 Picket fence effect
Fig,2.73 The DFT of a periodic signal without full-period truncation
Conclusion:
Full-period truncation of a periodic signal
is a prerequisite for obtaining the correct
discrete spectrum.
2.3.5 Picket fence effect
Measurement Technology
(4)
Prof,Wang Boxiong
Digital signal processing,a field which has
its roots in 17th and 18th century mathematics,
has become an important modern tool in a
multitude of diverse fields of science and
technology,
Digital signal processing is concerned with
the representation of signals by sequences
of numbers or symbols and the processing
of these sequences,
2.3 Digital signal processing
The availability of high speed digital
computers has fostered the development of
increasingly complex and sophisticated signal
processing algorithms,and recent advances
in integrated circuit technology promise
economical implementations of very complex
digital processing systems,
The evolution of a new point of view toward
digital signal processing was further
accelerated by the disclosure in 1965 of an
efficient algorithm for computation of Fourier
transforms,the fast Fourier transform or FFT,
2.3 Digital signal processing
The fast Fourier transform algorithm
reduced the computation time of Fourier
transform by orders of magnitude,
The importance of digital signal processing
appears to be increasing with no visible sign
of saturation,
The impact of digital signal processing
techniques will undoubtedly promote
revolutionary advances in some fields of
application,
2.3 Digital signal processing
For a nonperiodic continuous time signal
x(t),its Fourier transform must be a
continuous spectrum X(f),
The continuous time signals and the
continuous spectra must be discretized first
and then truncated to get a finite length of
sequence before being processed by a
computer,This forms just the basis for the
discrete Fourier transform (DFT),
2.3.1 Discrete Fourier Transform (DFT)
dtetxfXFT ftj?2,(2.199)
dfefXtxI F T ftj?2, (2.200)
There are four cases for the Fourier transform
of an infinite-length continuous signal (Fig,
2.63),
2.3.1 Discrete Fourier Transform (DFT)
Fig,2.63 Types of Fourier transform
Fig,2.63 (a),a nonperiodic continuous
signal x(t) and its Fourier transform
spectrum X(f),The spectrum is continuous.
Fig,2.63 (b),a periodic continuous signal,
and the frequency spectrum is or discrete,
where
Δf,fundamental frequency,
2.3,1 Discrete Fourier Transform (DFT)
2
2
21:
T
T
ktfj
k dtetxTfXFT
(2.201)
k
tfj
k kefXtxI F T
2:
(2.202)
),2,1,0( kfkf k
Tf
1
Fig,2.63 (c),the Fourier transform of a
nonperiodic discrete signal,The Fourier
transform of an infinite-length discrete time
sequence is a periodic continuous spectrum,
where
Δt is the sampling period; fs is the
sampling frequency of the time sequence.
2.3.1 Discrete Fourier Transform (DFT)
n
tfj
n netxfXFT
2:
(2.203)
2
2
21,s
s
n
f
f
tfj
s
n dfefXftxI F T
(2.204)
),2,1,0( ntnt n
sf
t 1
Fig,2.63 (d),the Fourier transform of a
periodic discrete time sequence,Its
spectrum is also periodic and discrete,
The sampling period is Δt,then
2.3.1 Discrete Fourier Transform (DFT)
tNT
Conclusion:
For a periodic x(t) the spectrum X(f) is
bound to be discrete,and vice versa,
If x(t) is nonperiodic,then X(f) is
continuous,and vice versa,
The case shown in Fig,2.64 (d) where
both the time and frequency signals are
discrete and periodic provides us with the
possibility of using a computer to
implement spectrum analysis,
2.3.1 Discrete Fourier Transform (DFT)
DFT,a Fourier representation of a finite-
length sequence which itself is a sequence
rather than a continuous function,and it
corresponds to samples equally spaced in
frequency of the Fourier transform of the
signal,
where,x(n) and X(k) are periods for
and respectively,and Δt and f0 are
generalized to be unity,
2.3.1 Discrete Fourier Transform (DFT)
1,,1,0,11 2
NkWnxenxkXD FT nk
N
N
on
N
on
nkNj(2.205)
1,,1,011,1 12
NnWkX
NekXNnxI D FT
N
oK
N
oK
nk
N
nkNj(2.206)
NjN eW?2 )(? tnx?
)(? 0kfX
The true meaning of the DFT,
– It is possible to sample and truncate any
continuous time signals and make DFT of it to
get a discrete spectrum,whose envelope is the
estimate of the true spectrum of the original
continuous signal,
DFT process:
– sampling in time domain;
– truncating in time domain;
– sampling in frequency domain,
2.3.1 Discrete Fourier Transform (DFT)
2.3.1 Discrete Fourier Transform (DFT)
Fig,2.64 Graphical representation of the DFT (a)
2.3.1 Discrete Fourier Transform (DFT)
Fig,2.64 Graphical representation of the DFT (b)
2.3.1 Discrete Fourier Transform (DFT)
Fig,2.64 Graphical representation of the DFT (c)
2.3.1 Discrete Fourier Transform (DFT)
Fig,2.64 Graphical representation of the DFT (d)
1,Linearity
If
then
where a and b are constants,
2.3.2 Properties of the discrete Fourier transform
)()]([
)()]([
22
11
kXnxD F T
kXnxD F T
kbXkaXnbxnaxD F T 2121 (2.214)
2,Shift of a sequence
If
time shifting:
frequency shifting:
2.3.2 Properties of the discrete Fourier transform
)()]([ kXnxD F T?
NkmjekXmnxD F T /2][ (2.215)
mkXenxD F T Nkmj ][ /2? (2.216)
3,Symmetry property
1) If x(n) is a complex sequence,and
then
2) If x(n) is a real sequence,and
then
where XR(k) and XI(k) are the real part and
imaginary part of X(k) respectively,
2.3.2 Properties of the discrete Fourier transform
)()]([ kXnxD F T?
kXnxD FT ** (2.217)
)()]([ kXnxD F T?
kXkX
kNXkX
kNXkXkX
kNXkXkX
kNXkXkX
III
RRR
a r ga r g
*
(2.218)
3) If x(n) is an even sequence,that is,
x(n)=x(-n),then X(k) is a real sequence,
4) If x(n) is an odd sequence,that is,x(n)=-
x(-n),then X(k) is a pure imaginary
sequence,
2.3.2 Properties of the discrete Fourier transform
4,Parseval’s theorem
5,Convolution
If
then
2.3.2 Properties of the discrete Fourier transform
1 1 22 1N
on
N
ok
kXNnx (2.219)
)()]([
)()]([
kYnyD F T
kXnxD F T
kYkXnynx*
(2.220)
kYkXnynx * (2.221)
If the sampling rate is too low or the
sampling interval is too large,then the
resulted discrete time sequence may not
represent correctly the waveform of the
original signal,and frequency confusion
or aliasing will occur in frequency domain
processing,
2.3.3 Sampling theorem
2.3.3 Sampling theorem
Fig,2.65 Effect of varying the sample rate,fs,on the apparent
signal obtained by discrete sampling
2.3.3 Sampling theorem
Fig,2.65 Effect of varying the sample rate,fs,on the apparent
signal obtained by discrete sampling (continued)
The highest frequency resolved at a given
sampling frequency is determined by the
Nyquist frequency,and
To prevent these problems,the sampling
frequency should always be chosen to be
more than twice a signal’s highest frequency,
Eq,(2.223) is the sampling theorem,
2.3.3 Sampling theorem
2
s
Nyq
ff? (2.222)
ma x2 ff s?
(2.223)
2.3.3 Sampling theorem
Fig,2.66 Conditions for aliasing
(a) With frequency aliasing,when fs<2fmax
2.3.3 Sampling theorem
Fig,2.66 Conditions for aliasing
(b) Without frequency aliasing,when fs>2fmax
Windowing,to suppress or reduce leakage,
special window functions of better properties
are selected.
2.3.4 Leakage and windowing
Fig,2.67 Leakage caused by windowing for a sinusoidal function
The characteristic of a window
function:
1) 3dB-bandwidth B,the bandwidth for
which the generalized amplitude of the
main lobe remains within of its
maximum value,A generalized
amplitude of W(f):
2) Peak amplitude of side-lobe A (dB),the
ratio of the peak amplitude of side-lobe,
Asmax,to the main lobe amplitude,Am,
that is,
2.3.4 Leakage and windowing
)dB( 321?
0/lg20 WfWfW?
mm a xs AAlg20
3) Side-lobe attenuation (minimum stop-
band attenuation) D (dB/decade),the ratio
of the peak amplitude of side-lobe to the
amplitude of the side-lobe at a decade
frequency,
2.3.4 Leakage and windowing
Fig,2.68 Definition of parameters of a window function spectrum
Note:
An ideal window function must have
minimum B and minimum A as well
as maximum D,
2.3.4 Leakage and windowing
2.3.4 Leakage and windowing
Fig,2.69 Several commonly-used windows
2.3.4 Leakage and windowing
Fig,2.70 Fourier transform of windows of Fig,2.69
Picket fence effect,If the frequency fi of
a harmonic component in a signal is equal
to k/T,that is,if it coincides with the
sampling point of the output,then this
spectral line will be accurately displayed,
On the contrary,if fi does not coincide with
the sampling point,the spectral line for fi
cannot be displayed and this leads to an
error with the spectrum.
2.3.5 Picket fence effect
Frequency resolution Δf,the spacing
between two neighboring spectral lines.
When the length T of a time signal (the
window width T=NTs) and the sampling
frequency fs are fixed,so is the frequency
resolution Δf,
2.3.5 Picket fence effect
TN
ff s 1 (2.231)
2.3.5 Picket fence effect
Fig,2.72 The DFT of a periodic signal with full-period truncation
2.3.5 Picket fence effect
Fig,2.72 The DFT of a periodic signal with full-period truncation (continued)
2.3.5 Picket fence effect
Fig,2.73 The DFT of a periodic signal without full-period truncation
Conclusion:
Full-period truncation of a periodic signal
is a prerequisite for obtaining the correct
discrete spectrum.
2.3.5 Picket fence effect
