Download Fourier Analysis and Signal Processing: Understanding Analog and Digital Signals and more Summaries Digital Signal Processing in PDF only on Docsity!
Slides adapted from ME Angoletta, CERN
sampling time, t
k
[ms]
Voltage [V]
ts
sampling time, t
k
[ms]
Voltage [V]
ts
Analog & digital signalsAnalog & digital signals
Continuous function^ Continuous function
V of
continuous^ continuous
variable t (time,
space etc) : V(t).
Analog
Discrete function^ Discrete function
V
k
of
discrete^ discrete
sampling variable t
k
with k = integer: V
k =
V(t
k
Digital
time [ms]
Voltage [V]
Uniform (periodic) sampling.Sampling frequency f
S
= 1/ t
S
Slides adapted from ME Angoletta, CERN
Digital
vs
analog proc’ing
Digital
vs
analog proc’ing
Digital Signal Processing (DSPing)
- More flexible.• Often easier system upgrade.• Data easily stored.• Better control over accuracy
requirements.
AdvantagesAdvantages
- A/D & signal processors speed:
wide-band signals still difficult totreat (real-time systems).
- Finite word-length effect.• Obsolescence (analog
electronics has it, too!).
LimitationsLimitations
Slides adapted from ME Angoletta, CERN
Digital system implementationDigital system implementation
• Pass / stop bands. • Sampling rate.
KEY DECISION POINTS:^ KEY DECISION POINTS:
Analysis bandwidth, Dynamic range
• No. of bits. Parameters.
Digital
Processing
A/D
Antialiasing
Filter
ANALOG INPUTANALOG INPUT DIGITAL OUTPUTDIGITAL OUTPUT
• Digital format.
What to use for processing?See slide “DSPing aim & tools”
Slides adapted from ME Angoletta, CERN
SamplingSampling
How fast must we sample
a continuous
signal to preserve its info content?
Ex: train wheels in a movie. 25 frames (=samples) per second.
Frequency misidentification due to low sampling frequency.
Train starts
wheels ‘go’ clockwise.
Train accelerates
wheels ‘go’ counter-clockwise.
Why?Why?
Sampling: independent variable (ex: time) continuous
→
discrete.
Quantisation: dependent variable (ex: voltage) continuous
→
discrete.
Here we’ll talk about uniform sampling
.
*^ *
Slides adapted from ME Angoletta, CERN
The sampling theoremThe sampling theorem
A signal s(t) with maximum frequency f
MAX
can be
recovered if sampled at frequency
f
S
> 2 f
MAX
Condition on f
S
?
f
S
300 Hz
t)
cos(
π
t) π
sin(
10
t)
π
cos(
3
s(t)
− ⋅ + ⋅ = F
1
=25 Hz, F
2
= 150 Hz, F
3
= 50 Hz
F
F
F
f
MAX
Example
Theo
Multiple proposers: Whittaker(s), Nyquist, Shannon, Kotel’nikov.
Nyquist frequency (rate) f
N
= 2 f
MAX
or
f
MAX
or
f
S,MIN
or
f
S,MIN
/
Naming getsconfusing!
Frequency domain Slides adapted from ME Angoletta, CERN
(hints)
Frequency domain
(hints)
-^ •
Time & frequencyTime & frequency
: two complementary signal descriptions.
Signals seen as “projected’ onto time or frequency domains.
-^ •
BandwidthBandwidth
: indicates rate of change of a signal.
High bandwidth
signal changes fast.
Ear^ Ear
- brain act as frequency analyser: audio spectrum
split into many narrow bands
low-power sounds
detected out of loud background.
Example
Slides adapted from ME Angoletta, CERN
Antialiasing filterAntialiasing filter
-B
0
B
f
Signal of interest
Out of band
noise
Out of band
noise
-B
0
B f
S
/
f
(a),(b)
Out-of-band
noise can alias
into band of interest. Filter it before!
Filter it before!
(a)
(b)
-B
0
B
f
Antialiasing
filter
(c) Passbandfrequency
Passband
: depends on bandwidth of
interest. Attenuation A
MIN
: depends on
- ADC resolution ( number of bits N).
A
MIN, dB
~ 6.02 N + 1.
- Out-of-band noise magnitude.Other parameters: ripple, stopbandfrequency...
(c)
AntialiasingAntialiasing filter
filter
Slides adapted from ME Angoletta, CERN
ADC - Number of bits NADC - Number of bits N
Continuous input signal digitized into
N
levels
(^3210) -1 -2 -3 -
**-
-**
0
1
2
3
4
(^111010001000)
V
V
FSR
Uniform, bipolar transfer function (N=3)Uniform, bipolar transfer function (N=3)
Quantization step q =^ Quantization step
V
FSR 2
N
Ex: V
FSR
= 1V , N = 12
q = 244.
V
LSBLSB
Voltage ( = q)Scale factor (= 1 / 2
N
)
Percentage (= 100 / 2
N
)
- -0.
0 0.
1
**-
-**
0
1
2
3
4
q / 2 - q / 2
Quantisation errorQuantisation error
Slides adapted from ME Angoletta, CERN
Frequency analysis: why?Frequency analysis: why?
Fast & efficient insight on signal’s building blocks.
Simplifies original problem - ex.: solving Part. Diff. Eqns. (PDE).
Powerful & complementary to time domain analysis techniques.
The brain does it?
time, t
frequency, f
F
s(t)
S(f) =
F
[s(t)]
analysisanalysis
synthesissynthesis
s(t), S(f) :
Transform Pair
General Transform asGeneral Transform as
problemproblem-
-solving tool
solving tool
Slides adapted from ME Angoletta, CERN
Fourier analysis - toolsFourier analysis - tools
Input Time Signal
Frequency spectrum
N
n
N
n k π 2 j
e
s[n]
1 N
k ~c
Discrete
Discrete
DFSDFS
Periodic (period T)
Continuous
DTFT
Aperiodic
Discrete
DFTDFT
n f π 2 j
e
n
s[n]
S(f)
2.5^2 1.5^1 0.5^00
2
4
6
8
10
12
time, t
k
2.5^2 1.5^1 0.5^00
1
2
3
4
5
6
7
8
time, t
k
N
n
N
n k π 2 j
e
s[n]
1 N
k ~c
Calculated via FFT
dt t f π j
e
s(t)
S(f)
dt
T^0
t ω kj
e
s(t)
1 T
k c
Periodic (period T)
DiscreteContinuous
FTFT
Aperiodic
FSFS
Continuous
2.5^2 1.5^1 0.5^00
1
2
3
4
5
6
7
8
time, t
2.5^2 1.5^1 0.5^00
2
4
6
8
10
12
time, t
Note: j =
√
-1,
ω
= 2
π
/T, s[n]=s(t
n
), N = No. of samples
Slides adapted from ME Angoletta, CERN
A little history -2A little history -
¾¾
(^1919)
thth
/ 20/ 20
thth
centurycentury
: two paths for Fourier analysis
two paths for Fourier analysis -
Continuous & Discrete.
CONTINUOUSCONTINUOUS →
Fourier extends the analysis to arbitrary function (Fourier Transform).
→
Dirichlet, Poisson, Riemann, Lebesgue address FS convergence.
→
Other FT variants born from varied needs (ex.: Short Time FT - speech analysis).
DISCRETE: Fast calculation methods (FFT)DISCRETE: Fast calculation methods (FFT) →
18051805
- Gauss, first usage of FFT (manuscript in Latin went unnoticed!!!
Published 1866).
→
19651965
- IBM’s Cooley & Tukey “rediscover” FFT algorithm (
“An algorithm for
the machine calculation of complex Fourier series”
).
→
Other DFT variants for different applications (ex.: Warped DFT - filter design &signal compression).
→
FFT algorithm refined & modified for most computer platforms.
Fourier Series (FS)Fourier Series (FS) Slides adapted from ME Angoletta, CERN
see next slidesee next slide
A periodic^ A
periodic function s(t) satisfying
function s(t) satisfying Dirichlet
Dirichlet’
’ss conditions
conditions
can be expressedcan be expressed
as a Fourier series, with harmonically related sine/cosine terms.as a Fourier series, with harmonically related sine/cosine terms
.
[
]
∞
=
⋅
−
⋅
=
1
k
t)
ω
(k
sin
k b
t)
ω
(k
cos
k a
0 a
s(t)
a
0
, a
k
, b
k
: Fourier coefficients.
k
: harmonic number, T
: period,
ω
= 2
π
/T
For all t but discontinuitiesFor all t but discontinuities
Note: {cos(k
ω
t), sin(k
ω
t) }
k
form orthogonal base offunction space.
⋅
=
T 0
s(t)dt
(^1) T
0 a
⋅
⋅
=
T^0
dt t)
ω
sin(k
s(t)
(^2) T
k b
⋅
⋅
=
T 0
dt t)
ω
cos(k
s(t)
(^2) T
k a
(signal average over a period, i.e. DC term &zero-frequency component.)
analysis analysis
synthesissynthesis
Slides adapted from ME Angoletta, CERN
FS analysis - 1FS analysis - 1
*** Even & Odd functions Odd :**
s(-x) = -s(x)
x
s(x) s(x)
x
Even :
s(-x) = s(x)
FS of odd
function:
square wave.
-1.
- -0.
0 0.
1 1.
0
2
4
6
8
10
t
square signal, sw(t)
2 π
π^0
π π
1)dt (
dt
1 π
a
∫
∫
π^0
π π
dt kt
cos
dt kt
cos
(^1) π
k a
∫
∫
{
}
∫
∫
k
π
cos
1
π k
π^0
π π
dt kt
sin
dt kt
sin
(^1) π
k b
even k ,
0
odd k ,
π k
ω
π
T
(zero average)^ (zero average)
(odd function)(odd function)
...
t
5
sin
π
5
4
t
3
sin
π
3
4
t
sin
(^4) π
sw(t)
Slides adapted from ME Angoletta, CERN
[
]
=
⋅
=
7
1
k
sin(kt)
k b
(t) 7
sw
-1.
- -0.
0 0.
1 1.
0
2
4
6
8
10
t
square signal, sw(t)
-1.
- -0.
0 0.
1 1.
0
2
4
6
8
10
t
square signal, sw(t)
[
]
=
⋅
=
5
1
k
sin(kt)
k b
(t) 5
sw
[
]
=
⋅
=
3
1
k
sin(kt)
k b
(t) 3
sw
-1.
- -0.
0 0.
1 1.
0
2
4
6
8
10
t
square signal, sw(t)
[
]
=
⋅
=
1
1
k
sin(kt)
k b
(t) 1
sw
-1.
- -0.
0 0.
1 1.
0
2
4
6
8
10
t
square signal, sw(t)
-1.
- -0.
0 0.
1 1.
0
2
4
6
8
10
t
square signal, sw(t)
[
]
=
⋅
=
9
1
k
sin(kt)
k b
(t) 9
sw
-1.
- -0.
0 0.
1 1.
0
2
4
6
8
10
t
square signal, sw(t)
-1.
- -0.
0 0.
1 1.
0
2
4
6
8
10
t
square signal, sw(t)
[
]
=
⋅
=
11
1
k
sin(kt)
k b
(t)
11
sw
FS synthesisFS synthesis
Square wave reconstructionSquare wave reconstruction
from spectral termsfrom spectral terms
Convergence may be slow (~1/k) - ideally need infinite terms. Practically
, series truncated when remainder below computer tolerance
(
⇒
errorerror
).
BUT^ BUT
… Gibbs’ Phenomenon.
