Introduction to
Digital Signal Processing
(Discrete-time Signal Processing)
Prof. Chu-Song Chen
Research Center for Info. Tech. Innovation,
Academia Sinica, Taiwan
Dept. CSIE & GINM
National Taiwan University
Fall 2012
Course Information
• Teaching assistant:
– Yin-Tzu Lin 林映孜
known@cmlab.csie.ntu.edu.tw
• Course webpage
– www.cmlab.csie.ntu.edu.tw/~dsp/dsp2012
• Grades
– Homework x 2 (30%)
– Test x 2 (40%)
– Term project (30%)
Review of Complex Exponential
Real and Image Parts:
Phase Difference is /2
Analogous to Uniform Circular Motion
Time axis
Advantage in computation
Basis Functions
• Remember that, sinusoids (such as sine) can serve
as orthogonal bases for representing signals.
• But, why not using sinusoids directly as basis
functions in signals and linear systems?
Basis Functions in Signals and Systems
• In signal processing, we hope that each basis function
corresponds to a frequency, so that a signal can be
decomposed into different frequency components.
• However, when the frequency is chosen, there are still
various phases φ corresponding to this frequency.
cos(w0t+φ)
• Eg., when the frequency is fixed to be w0 in a cosine
function, what is the phase φ that should be chosen to
serve as a basis?
• Otherwise, do we need to use all the phases as basis
functions? (seems to be redundant)
Complex Exponentials
Serving as Basis Functions
• Linear combination:
 coefficients  Bases
• Can we simply use zero-phase functions as bases?
• This cannot be achieved easily by sinusoid.
• For example, if we use zero-phase sine functions
{sin(t) | R} as bases, the functions f we can
represent is restrict to those satisfies f(0)=0,
because sin(0)=0 and so the linear combination
of zero-phase sine functions is always zero.
Why choosing complex exponential?
• Nevertheless, it can be shown that we can use
ejt as bases. That is, ej(t+) when zero phase
(=0).
• In this way, we can ignore the phase and focuses
only on frequency in the bases representation.
• Hence, we can decompose a signal into
components of different frequencies by using
complex exponentials (under some
requirements), instead of a more redundant
representation of different frequency–phase
pairs.
Orthogonality of complex exponentials
• First, the zero-phase complex exponential ejt can
form orthogonal bases.
• For simplicity, we take Fourier series as an
example, which deals with only periodic signals.
– Consider the functional space consisting of all of the
periodical signals with period T0, i.e., x(t) belongs to
this space if x(t)=x(t+T0).
• Let the k-th basis vk(t) in Foruier series be
which corresponds to the frequency
Orthogonality of complex exponentials
• Then, vk(t), which forms orthogonal bases:
where * denotes the complex conjugate. That is,
Z=a+bj, then z* = a-bj. When z is real, z=z*.
• Remark: please be reminded that in complex
variables, the inner product of two vectors x and y is
denoted as (x*)Ty, not xTy.
Proof of the orthogonalility of vk(t)
• Preliminary property:
• proof
1 at
1 at
– In real calculus,  e dt   e d (at )  e
a
a
where a is a real number.
– In complex variables, the above equalities still
hold when a is a complex number.
at
Proof of the orthogonalility of vk(t)
• Hence, we have
Proof of the orthogonalility of vk(t)
• Now, let us consider
Proof of the orthogonalility of vk(t)
• When k  l, by the preliminary property, we
have
Proof of the orthogonalility of vk(t)
• When k = l,
Proof of the orthogonalility of vk(t)
• Hence, we conclude that
• This shows the validity that zero-phase complex
exponential can serve potentially as basis functions.
• Remark: if you are not familiar with Fourier series, don’t
worry and we will introduce in details it later.
Phasors
Complex number coefficients
• Phasors = Complex Amplitude
• In addition to orthogonal bases property, there
is another requirement to employ zero-phase
complex exponentials as bases functions:
• The coefficients to be used in the linear
combination are extended as complex
numbers (instead of real numbers only).
• Remember that
j  w0t  
jw0t
j
Ae
 ( Ae )e
Phasors
Complex number coefficients
• That is, both amplification and phase-shift can
be represented as a complex-number
coefficient or complex amplitude (which is
multiplied to the basis)
Ae
j  w0t  
j
 ( Ae )e
Both amplitude and phase
are multiplied to the basis
jw0t
Zero-phase basis
Summary of the issue of bases
• In sum, zero-phase complex exponentials are
suitable to serve as bases for signal
representations because they satisfy
– Orthogonality property
– Allow the use of complex-number coefficients (complex
amplitude) to represent both amplitude and phase.
• In signals and systems, complex exponentials are
standard for serving as (frequency-related) basis
functions.
• However, in other problem domains, such as
compression and feature extraction, real-value
bases such as cosine functions or wavelet
functions are often more suitable.
More on Calculation Tricks of
Complex Exponentials
More on Calculation Tricks of
Complex Exponentials
Complex Amplitude
Sum of Sinusoids Calculation
Sum of sinusoids of the equal frequencies is still a
sinusoid of the same frequency.
How?
Amp
Phase
Illustration Example
Adding Sinusoids (Calcluation)
Adding Sinusoids
Example
Adding Sinusoids
Adding Sinusoids
Another example
Spectrum Representation
• Frequency-domain representation of a signal.
• As we know, a signal can be decomposed as a
linear combination of zero-phase complexexponential basis functions.
• When doing such a decomposition, the
coefficients obtained are referred to as the
spectrum of the signal.
Spectrum of a single sinusoid
• What is the spectrum of a single cosine function?
• Note that we employ complex exponential as
bases.
• Since
the spectrum is (,1/2), (, 1/2), containing
both positive and negative frequencies:
1/2



Example
• Even summing the complex exponentials, we
still get a real-value signal
Example:
Spectrum Representation
• The most straightforward way of viewing and
understanding a spectrum is to producing new
signals from sinusoids by additive linear
combination,
where a signal is created by adding together a
constant and N sinusoids of different
frequencies:
Spectrum Representation
• By the inverse Euler formula
• It gives a way to represent x(t) in the
alternative form:
Spectrum Representation
• We define the two-sided spectrum of a signal
x(t) composed of sinusoids to be the set of
2N+1 complex amplitudes corresponding to
the 2N+1 frequencies:
• We term it as the frequency-domain
representation of x(t).
Example
• Apply the inverse Euler formula
• The spectrum:
Example
• They are called the frequency components.
DC Component
• The constant component (ie., corresponding the zero
frequency) is referred to as the DC component.
• In the above example, the DC component is 10.
• We can separate the frequency components into the
amplitude (magnitude) and phase components.
Example: Synthetic Sound
• A periodic signal can be synthesized as the sum
complex exponentials
• How is it sounds like: consider a signal containing
nonzero terms for only
Vowel Example: Single component a2
Vowel: Two components: a2+a4
Vowel: Three components: a2+a4+a5
Vowel: Four components:
a2+a4+a5+a16
Five components: a2+a4+a5+a16+a17
Vowel signal: Frequency Domain
Multiplication of Sinusoids
Beat note and amplitude modulation (AM)
• When we multiply two sinusoids have
different frequencies, we can create an
interesting audio effect called a beat note.
• Another use for multiplying sinusoids is
modulation for radio broadcasting. AM radio
stations use this method.
• Multiplication of two sinusoids can be
equivalently represented as sum of two
sinusoids, as shown below.
Example of Multiplication of Sinusoids
Product of
two sinusoids
• By inverse Euler formula,
Sum of two sinusoids
General Derivation
Sum of two sinusoids
• Let
and
Product of two sinusoids
In the Spectrum Domain
• Spectrum of the multiplication of two
sinusoids of frequencies f = (f2-f1)/2 and fc =
(f2+f1)/2
• So, f2 = fc + f , f1 = fc - f
Example: Beat Note Signal
Envelope
effect
Amplitude Modulation (AM) Signal
• The AM signal is is a product of the form
where it is assumed that the frequency of the
cosine term (fc Hz) is much higher than any
frequencies contained in the spectrum of v(t).
• Example: let v(t)=5+4cos(40t) and fc =200Hz, its
waveform is shown in the following.
– unlike the beat note example, we have a DC
component here and so the envelope never goes to
zero)
Example: AM Signal
Envelope
Frequency Domain
Spectrum of the AM Signal
Periodic Waveforms
• A periodic signal satisfies the condition that
x(t+T0)=x(t) for all t.
• The time interval T0 is called the period of x(t).
• If it is the smallest such repetition interval, it is
called the fundamental period.
• Harmonically related frequencies: all
frequencies are integer multiples of a
frequency f0.
Periodic Waveforms
• The signal would be synthesized as the sum of N+1
cosine waves with harmonically related frequencies:
• The frequency fk is called the k-th harmonic of f0, the
fundamental frequency.
• When we add harmonically related complex
exponentials, we get a periodic signal.
Nonperiodic Waveforms
• What happen when the frequencies have no
harmonic relation to one another?
• If there are no harmonic assumptions on the
individual frequencies fk.
• When the combination frequencies are nonharmonic (i.e., does not have a fundamental
frequency), the signal could not be periodic.
Example: Nonperiodic Waveforms
obtained by summing sinusoids
• Sum of three cosine waves with nonharmonic
frequencies:
Fourier Series
• Any periodic signal can be synthesized by
sum of harmonically related sinusoids.
• The sum may need a infinite number of terms.
• This is the mathematical theory of Fourier
series:
Fourier Series
• How do we derive the coefficients ak?
• Remember that we have shown that the Fourier
series bases
have the
orthogonality property:
• Hence, to derive ak, we need simply to project
the signal x(t) onto the orthogonal basis vk(t) by
inner product with proper normalization.
Fourier Series
• Obtain ak by inner product of x(t) and vk(t):
• In particular, from the above the DC component
is obtained by
• That is, a0 is simply the average value of the
signal over one period.
Derivation Details
Transform pair of Fourier Series
• In sum, we have the following transform pair that
can be used for the analysis of periodic signals:
transform
pair
• The left, from x(t) to ak, is called the forward
transform, which transform the signal x(t) to the
frequency domain, and ak are frequencies or
spectrum.
• The right, from ak to x(t), is called the inverse
transform.
Sound Example of Periodic Signals:
Sine Wave
Sound Example of Periodic Signals:
Square Wave
Sound Example of Periodic Signals:
SAW Wave
Sound Example of Beats Wave