p. 1DSP-II
ALARI/DSP
INTRODUCTION-1
Toon van Waterschoot & Marc Moonen
Dept. E.E./ESAT, K.U.Leuven
http://homes.esat.kuleuven.be/~tvanwate
ALARI/DSP May 2013 p. 2 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
INTRODUCTION-1 : Who we are
• KU Leuven, Belgium– Dept. of Electrical Engineering (ESAT): signal & system
theory, micro- and nano-electronics, telecommunications, electrical energy, computer & document architecture, speech and image processing, …• SCD (SISTA-COSIC-DOCARCH): system identification, signal
processing, bio-informatics, cryptography, linear algebra, …– DSP (Digital Signal Processing): digital audio and
communications• Research topics: acoustic echo and feedback
cancellation, acoustic noise reduction, dereverberation, multicarrier communication, channel equalization, …
• Applications: hearing aids, public address systems, ADSL, wireless communication systems, …
ALARI/DSP May 2013 p. 3 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
INTRODUCTION-1 : Course schedule
• Monday – 14h00 - 17h00: Introduction – Questions & Answers (Toon van Waterschoot)
• Tuesday – 9h00 – 10h30: Lecture-1 (Marc Moonen) – 11h00 - 13h00: Exercise Session-1 (Toon van Waterschoot) – 14h00 – 15h30: Lecture-2 (Marc Moonen) – 16h00 - 19h00: Exercise Session-2 (Toon van Waterschoot)
• Wednesday – 9h00 – 10h30: Lecture-3 (Marc Moonen) – 11h00 - 13h00: Exercise Session-3 (Toon van Waterschoot) – 14h00 – 15h30: Lecture-4 (Marc Moonen) – 16h00 - 19h00: Exercise Session-4 (Toon van Waterschoot)
• Thursday – 9h00 – 10h30: Lecture-5 (Marc Moonen) – 11h00 - 13h00: Exercise Session-5 (Toon van Waterschoot) – 14h00 – 15h30: Lecture-6 (Marc Moonen) – 16h00 - 19h00: Exercise Session-6 (Toon van Waterschoot)
• Friday – 9h00 – 10h30: Lecture-7 (Marc Moonen) – 11h00 - 13h00: Exercise Session-7 (Toon van Waterschoot) – 14h00 – 15h30: Lecture-8 (Marc Moonen)
ALARI/DSP May 2013 p. 4 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Course webpage:
http://homes.esat.kuleuven.be/~tvanwate/alari.html
INTRODUCTION-1 : Course webpage
ALARI/DSP May 2013 p. 5 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
INTRODUCTION-1 : Overview
• Introduction• Discrete-time signals
sampling, quantization, reconstruction• Stochastic signal theory
deterministic & random signals, (auto-)correlation functions, power spectra, …
• Discrete-time systemsLTI, impulse response, FIR/IIR, causality & stability,
convolution & filtering, …• Complex number theory
complex numbers, complex plane, complex sinusoids, circular motion, sinusoidal motion, …
ALARI/DSP May 2013 p. 6 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
INTRODUCTION-2 : Overview
• z-transform and Fourier transformregion of convergence, causality & stability, properties,
frequency spectrum, transfer function, pole-zero representation, …
• Elementary digital filtersshelving filters, presence filters, all-pass filters
• Discrete transformsDFT, FFT, properties, fast convolution,
overlap-add/overlap-save, …
ALARI/DSP May 2013 p. 7 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Introduction: overview
• Digital signal processing?• Analog vs. digital signal processing• Example: design of a delay audio effect
– in the analog world– in the digital world
ALARI/DSP May 2013 p. 8 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Introduction: digital signal processing?
Digital signal processing?
• Signal: a physical quantity which varies as a function of some independent variable(s)– 1-dimensional: sound signal (mechanical/electrical),
electromagnetic signal (wired/wireless), chemical concentration, …
– 2-dimensional: image– …– N-dimensional: …
• Independent variable: time, position, frequency, …
her
e…
ALARI/DSP May 2013 p. 9 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Introduction: digital signal processing?
Digital signal processing?
• Processing: altering the signal characteristics to improve signal quality– equalization: to undo the (frequency-selective) effect
of passing the signal through a system (channel)– noise reduction: to remove noise/interference– signal separation: to separate multiple signals which
are present in one measurement– modulation: to prepare a signal for being transmitted
through a frequency-selective channel– …
• Processing ~ Filtering
ALARI/DSP May 2013 p. 10 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Introduction: digital signal processing?
Digital signal processing?
• Digital: the signal processing is performed by a finite number of operations using a finite number of digits– discretization of independent variable: the signal is
sampled w.r.t. the (continuous) independent variable (e.g., discrete time, discrete frequency, …)
– discretization of signal value: the signal value (amplitude) is approximated on a discrete scale (quantization)
• Bits: digital signals are often represented using binary digits = bits
ALARI/DSP May 2013 p. 11 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Introduction: analog vs. digital SP
Analog electrical
signal processing
circuit
Analog world
Analog signal processing: “how things used to be”
Analog IN Analog OUT
ALARI/DSP May 2013 p. 12 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Introduction: analog vs. digital SP
Analog-to-
digital
conversionDSP
Digital-to-
analog
conversion
Analog world Analog worldDigital world
Analog IN Analog OUTDigital IN
Digital OUT
0110100101
1001100010
Digital signal processing in the analog world
ALARI/DSP May 2013 p. 13 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Introduction: analog vs. digital SP
• Analog world– Analog input: microphone voltage, satellite receiver
voltage, …
– Analog output: loudspeaker voltage, antenna voltage, …
VIN
0
VOUT
0
ALARI/DSP May 2013 p. 14 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Introduction: analog vs. digital SP
Analog-to-
digital
conversionDSP
Digital-to-
analog
conversion
Analog world Analog worldDigital world
Analog IN Analog OUTDigital IN
Digital OUT
0110100101
1001100010
Digital signal processing in the analog world
ALARI/DSP May 2013 p. 15 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Introduction: analog vs. digital SP
• Digital world– Digital signal processor (DSP): microprocessor
designed particularly for signal processing operations, incorporated in sound card, modem, mobile phone, mp3 player, digital camera, digital tv, hearing aid, …
ALARI/DSP May 2013 p. 16 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Introduction: design example
Example: design of a “delay” audio effect
• Goal: design and implement an audio effect which mixes a scaled and delayed version of an audio signal to the original signal
delay
operation
Analog IN Analog OUT
scaling
operation
mixing
operation
ALARI/DSP May 2013 p. 17 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Introduction: design example
Analog design:
Example: design of a “delay” audio effect
Analog IN Analog OUT
mixing
operation
delay
operationscaling
operation
ALARI/DSP May 2013 p. 18 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Introduction: design example
Digital design:
Example: design of a “delay” audio effect
Analog IN Analog OUTADC DAC
write new sample
buffer = {y[k], y[k-1], … y[k-D]}
read delayed sample
K*y[k-D]
y[k] = x[k] + K*y[k-D]
x[k]
inside
the DSP
mixing
operation
delay
operationscaling
operation
y[k]
ALARI/DSP May 2013 p. 19 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Introduction: design example
Example: design of a “delay” audio effect
• Analog design:– design of analog circuits– manufacturing of print board– assembly of analog components
• Digital design:– design of digital algorithm– compilation on digital signal processor
circuit design algorithm design
application-specific hardware re-usable hardware
ALARI/DSP May 2013 p. 20 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
INTRODUCTION-1 : Overview
• Introduction• Discrete-time signals
sampling, quantization, reconstruction• Stochastic signal theory
deterministic & random signals, (auto-)correlation functions, power spectra, …
• Discrete-time systemsLTI, impulse response, FIR/IIR, causality & stability,
convolution & filtering, …• Complex number theory
complex numbers, complex plane, complex sinusoids, circular motion, sinusoidal motion, …
ALARI/DSP May 2013 p. 21 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time signals: overview
• A/D conversion: sampling and quantization– time-domain sampling & spectrum replication– sampling theorem– anti-aliasing prefilters– quantization– oversampling and noise shaping
• D/A conversion: reconstruction– ideal vs. realistic reconstructors– anti-image postfilters
• Conclusion: DSP system block scheme
ALARI/DSP May 2013 p. 22
Discrete-time signals: sampling-quantization
Analog Signal
Processing Circuit
Analog Domain (Continuous-Time Domain)
Analog signal processing
Analog IN Analog OUT
)(tu
dtetufU tfj ..2).()(
(=Spectrum/Fourier Transform)
)(ty
dtetyfY tfj ..2).()(
Jose
ph
Fo
uri
er (
1768
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0)
Toon van Waterschoot & Marc Moonen INTRODUCTION-1
ALARI/DSP May 2013 p. 23 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time signals: sampling-quantization
Analog-to-
digital
conversionDSP
Digital-to-
analog
conversion
Analog world Analog worldDigital world
Analog IN Analog OUTDigital IN
Digital OUT
0110100101
1001100010
samplingquantization
ALARI/DSP May 2013 p. 24 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time signals: sampling
k
sD Tkttxtx ).().()(
• time-domain samplingamplitude amplitude
discrete-time [k]continuous-time (t)
impulse train
).(][ sTkxkx )(tx
sT
It will turn out (page 27) that a spectrum can be computed from x[k], which (remarkably) will be equal to the spectrum (Fourier transform) of the (continuous-time) sequence of impulses =
discrete-time
signal
continuous-time
signal
0 1 2 3 4
ALARI/DSP May 2013 p. 25 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time signals: sampling
• spectrum replication– time domain:
– frequency domain:
magnitude
frequency (f)
magnitude
frequency (f)
k
sD Tkttxtx ).().()(
k ss
D T
kfX
TfX )(.
1)(
)( fX )( fX D
ALARI/DSP May 2013 p. 26 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time signals: sampling
• sampling theorem– the analog signal spectrum has a bandwidth of fmax Hz
– the spectrum replicas are separated with fs =1/Ts Hz
– no spectral overlap if and only if
magnitude
frequency
ALARI/DSP May 2013 p. 27
• sampling theorem:– terminology:
• sampling frequency/rate fs
• Nyquist frequency fs/2
• sampling interval/period Ts
– e.g. CD audio: fmax ¼ 20 kHz ) fs = 44,1 kHz
Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time signals: sampling
• anti-aliasing prefilters:– if then frequencies above the Nyquist
frequency will be ‘folded back’ to lower frequencies
= aliasing– to avoid aliasing, the sampling operation is usually
preceded by a low-pass anti-aliasing filter
Har
ry N
yqui
st (
7 fe
brua
ri 18
89 –
4 a
pril
1976
)
ALARI/DSP May 2013 p. 28 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time signals: quantization
• B-bit quantization
amplitude
discrete time [k]
0Q
2Q3Q
-Q-2Q-3Q
R
)1on width quantizati
range(log bits ofnumber 2
Q
RB
amplitude
discrete time [k]
quantized discrete-time signal
=digital signaldiscrete-time signal
][kx ][kxQ
ALARI/DSP May 2013 p. 29 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time signals: quantization
• B-bit quantization:– the quantization error can
only take on values between and– hence can be considered as a random noise
signal with range – the signal-to-noise ratio (SNR) of the B-bit quantizer can
then be defined as the ratio of the signal range and the quantization noise range :
= the “6dB per bit” rule
ALARI/DSP May 2013 p. 30 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time signals: quantization
• oversampling:– it is possible to make a trade-off between sampling rate
and quantization noise– using a ‘coarse’ quantizer may be compensated by
sampling at a higher rate = oversampling– e.g. an increasing number of audio recordings is done at
a sampling rate of 96 kHz (while fmax ¼ 20 kHz )
• noise shaping:– the quantization noise is typically assumed to be white– the noise spectrum may be altered to decrease its
disturbing effect = noise shaping– e.g. psycho-acoustic noise shaping in audio quantizing
ALARI/DSP May 2013 p. 31 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time signals: reconstruction
Analog-to-
digital
conversionDSP
Digital-to-
analog
conversion
Analog world Analog worldDigital world
Analog IN Analog OUTDigital IN
Digital OUT
0110100101
1001100010
reconstruction
ALARI/DSP May 2013 p. 32 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time signals: reconstruction
• reconstructor: – ‘fill the gaps’ between adjacent samples– e.g. staircase reconstructor:
amplitude
discrete time [k]
amplitude
continuous time (t)
reconstructed
analog signal
discrete-time/digital signal
][kx )(txR
ALARI/DSP May 2013 p. 33 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time signals: reconstruction
• ideal reconstructor:– ideal (rectangular) low-pass filter– no distortion
magnitude
frequency
magnitude
frequency
• staircase reconstructor:– sync-like low-pass filter with sidelobes– distortion due to spurious high frequencies
magnitude
frequency
magnitude
frequency
)( fX D
)( fX D
ALARI/DSP May 2013 p. 34 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time signals: reconstruction
• anti-image postfilter:– low-pass filter to remove spurious high frequency
components due to imperfect reconstruction– comparable to the anti-aliasing prefilter
ALARI/DSP May 2013 p. 35 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time signals: conclusion
Digital OUT
x(t)
Analog IN
DSPDigital
IN
sampler quantizeranti-
aliasing prefilter
anti-image
postfilterreconstructor
Analog OUT
xp(t) x[k] xQ[k] y[k] yR(t)
y(t)
DSP system block scheme:
ALARI/DSP May 2013 p. 36 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
INTRODUCTION-1 : Overview
• Introduction• Discrete-time signals
sampling, quantization, reconstruction• Stochastic signal theory
deterministic & random signals, (auto-)correlation functions, power spectra, …
• Discrete-time systemsLTI, impulse response, FIR/IIR, causality & stability,
convolution & filtering, …• Complex number theory
complex numbers, complex plane, complex sinusoids, circular motion, sinusoidal motion, …
ALARI/DSP May 2013 p. 37 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Stochastic signal theory: overview
• Signal types:– deterministic signals– random signals
• Correlation functions and power spectra:– autocorrelation function & power spectrum– cross-correlation function & cross-spectrum– (joint) wide sense stationarity
• White noise:– Gaussian white noise– uniform white noise
ALARI/DSP May 2013 p. 38 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Stochastic signal theory: signal types
• Deterministic signals– a deterministic signal is an explicit function of time,
e.g. • Random signals
– a random signal is ‘unpredictable’ in a sense– some information on the signal behaviour may be
available, e.g.• probability density function (PDF) • mean • variance• autocorrelation function• …
ALARI/DSP May 2013 p. 39 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Stochastic signal theory: corr/spectra
• Autocorrelation function– measure of the dependence between successive
samples (with lag ) of a random signal
• Power spectrum– measure of the frequency content of a random signal– Fourier transform of the autocorrelation function
ALARI/DSP May 2013 p. 40 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Stochastic signal theory: corr/spectra
• Cross-correlation function– measure of the dependence between successive
samples (with lag ) of two different random signals
• Cross-spectrum– measure of spectral overlap between two random signals– Fourier transform of the cross-correlation function
ALARI/DSP May 2013 p. 41 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Stochastic signal theory: corr/spectra
• Wide-sense stationarity (WSS):– a random signal is wide-sense stationary if its mean and
autocorrelation function are independent of time:
• Joint wide-sense stationarity (joint WSS)– two random signals are jointly wide-sense stationary if
their cross-correlation function is independent of time:
ALARI/DSP May 2013 p. 42 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Stochastic signal theory: white noise
• White noise:– a zero-mean white noise signal has an impulse
autocorrelation function and a flat power spectrum:
– Gaussian white noise has a Gaussian PDF (Matlab function randn)
– uniform white noise has a uniform PDF (Matlab function rand)
power
time0
power
frequency0
ALARI/DSP May 2013 p. 43 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
INTRODUCTION-1 : Overview
• Introduction• Discrete-time signals
sampling, quantization, reconstruction• Stochastic signal theory
deterministic & random signals, (auto-)correlation functions, power spectra, …
• Discrete-time systemsLTI, impulse response, FIR/IIR, causality & stability,
convolution & filtering, …• Complex number theory
complex numbers, complex plane, complex sinusoids, circular motion, sinusoidal motion, …
ALARI/DSP May 2013 p. 44 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time systems: overview
• Introduction:– discrete-time systems– I/O behaviour
• LTI systems:– linear time-invariant systems– impulse response– FIR/IIR– causality– stability
• Convolution:– direct form– matrix form
ALARI/DSP May 2013 p. 45 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time systems: introduction
• discrete-time systems:– any system implemented on a digital signal processor:
– discrete-time model of continuous-time system, e.g.• wireless channel in mobile communications• twisted pair telephone line• acoustic echo channel between loudspeaker and microphone• …
DSPsampler quantizeranti-
aliasing prefilter
anti-image
postfilterreconstructor
xp(t) x[k] xQ[k] y[k] yR(t)
y(t)x(t)
ALARI/DSP May 2013 p. 46 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time systems: introduction
• input/output (I/O) behaviour:– mapping of input sequence on output sequence:
– the output signal is a function of the input signal:
systeminput output
ALARI/DSP May 2013 p. 47 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time systems: LTI systems
• Linear time-invariant (LTI) systems:– linearity:
– time-invariance:
ALARI/DSP May 2013 p. 48 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time systems: LTI systems
• Impulse response:– LTI systems are characterized uniquely by their impulse
response = the system output in response to a unit impulse input signal
amplitude
time
1
0time
1
amplitude
0
– the impulse response length – 1 is equal to the order of the system
ALARI/DSP May 2013 p. 49 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time systems: LTI systems
• Impulse response:– if the impulse response is known, the system response
to an arbitrary input signal can be calculated
amplitude
time
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
time
= + +
+ +=
ALARI/DSP May 2013 p. 50 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time systems: LTI systems
• FIR/IIR:– FIR: finite impulse response
– IIR: infinite impulse response
time
1
amplitude
0
time
1
amplitude
0
ALARI/DSP May 2013 p. 51 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time systems: LTI systems
• Causality:– a causal system has an impulse response that is zero
for all negative time indices– a non-causal system has an impulse response that has
some non-zero coefficients on the negative time axis, i.e. the system output depends on future input values
time
1
amplitude
0 time
1
amplitude
0
ALARI/DSP May 2013 p. 52 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time systems: LTI systems
• Stability:– a system is said to be stable if a bounded input signal
always generates a bounded output signal:
– a necessary and sufficient condition for stability is that the system impulse response be absolutely summable:
– instability can only occur with IIR systems
ALARI/DSP May 2013 p. 53 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time systems: convolution
amplitude
time
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
time
= + +
+ +=
ALARI/DSP May 2013 p. 54 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time systems: convolution
• Convolution:– the expression
can be written in a more general form:
– this operation is called convolution of the system impulse response with the input signal
– shorthand notation:
ALARI/DSP May 2013 p. 55 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time systems: convolution
• Convolution:
– if we define:• the impulse response length (with the system
order)• the input sequence length
– then the output sequence has length
ALARI/DSP May 2013 p. 56 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time systems: convolution
• Direct form convolution:– one way to perform the convolution of and
is to directly calculate the summation
– this is done for all time indices – an appropriate choice for the summation limits is:
ALARI/DSP May 2013 p. 57 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Discrete-time systems: convolution
• Matrix form convolution:– another way to perform the convolution of and
is by rewriting the summation as a matrix product
– the signal vectors and the impulse response matrix are defined as follows (with e.g. M=2 and L=4)
ALARI/DSP May 2013 p. 58 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
INTRODUCTION-1 : Overview
• Introduction• Discrete-time signals
sampling, quantization, reconstruction• Stochastic signal theory
deterministic & random signals, (auto-)correlation functions, power spectra, …
• Discrete-time systemsLTI, impulse response, FIR/IIR, causality & stability,
convolution & filtering, …• Complex number theory
complex numbers, complex plane, complex sinusoids, circular motion, sinusoidal motion, …
ALARI/DSP May 2013 p. 59 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Complex number theory: overview
• Complex numbers:– roots of a quadratic polynomial equation– fundamental theorem of algebra– complex numbers– complex plane
• Complex sinusoids– complex numbers complex sinusoids– circular motion– positive and negative frequencies– sinusoidal motion
ALARI/DSP May 2013 p. 60 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Complex number theory: complex numbers
complex numbers?
“imaginary” roots of a polynomial equation
ALARI/DSP May 2013 p. 61 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Complex number theory: complex numbers
• roots of a quadratic polynomial equation:– consider a quadratic polynomial, describing a parabola:
– the roots of the polynomial correspond to the points where the parabola crosses the horizontal -axis
ALARI/DSP May 2013 p. 62 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Complex number theory: complex numbers
• roots of a quadratic polynomial equation:– if the polynomial
has 2 real roots, and the parabola has 2 distinct intercepts with the -axis
– if the polynomial has 1 real root (with multiplicity 2), and the parabola has 1 intercept (tangent point) with the -axis
– if the polynomial has no real roots, and the parabola has no intercepts with the -axis
p(x)
xp(x)
xp(x)
x
ALARI/DSP May 2013 p. 63 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Complex number theory: complex numbers
• roots of a quadratic polynomial equation:– alternatively, if we could say that
the polynomial has 2 “imaginary roots”, and the parabola has 2 “imaginary” intercepts with the -axis
– these imaginary roots are represented as complex numbers:
with
p(x)
x
ALARI/DSP May 2013 p. 64 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Complex number theory: complex numbers
fundamental theorem of algebra:
every n-th order polynomial has exactly n complex roots
ALARI/DSP May 2013 p. 65 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Complex number theory: complex numbers
• complex numbers:– complex number:
– complex conjugate:
– modulus:
– argument:
• the complex numbers form a field, and all algebraic rules for real numbers also apply for complex numbers
ALARI/DSP May 2013 p. 66 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Complex number theory: complex numbers
• complex plane:– the modulus and argument naturally lead to a radial
representation in the complex plane
Im
Re
complex
plane
ALARI/DSP May 2013 p. 67 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Complex number theory: complex sinusoids
• complex variable complex sinusoid:
– from the radial representation we obtain
– replacing
– using Euler’s identity we get
ALARI/DSP May 2013 p. 68 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Complex number theory: complex sinusoids
• circular motion:– a complex sinusoid can be seen as a vector which
describes a circular trajectory in the z-plane
Im
Re
z-plane
ALARI/DSP May 2013 p. 69 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Complex number theory: complex sinusoids
• positive and negative frequencies:– for positive frequencies the circular motion is in
counterclockwise direction– for negative frequencies the circular motion is in
clockwise direction
Im
Re
Im
Re
ALARI/DSP May 2013 p. 70 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
Complex number theory: complex sinusoids
• sinusoidal motion:– sinusoidal motion is the projection of circular motion
onto any straight line in the z-plane, e.g.,• is the projection of onto the Re-axis• is the projection of onto the Im-axis
Im
Re
ALARI/DSP May 2013 p. 71 Toon van Waterschoot & Marc Moonen INTRODUCTION-1
INTRODUCTION-2 : Overview
• z-transform and Fourier transformregion of convergence, causality & stability, properties,
frequency spectrum, transfer function, pole-zero representation, …
• Elementary digital filtersshelving filters, presence filters, all-pass filters
• Discrete transformsDFT, FFT, properties, fast convolution,
overlap-add/overlap-save, …