EC320 Digital Signal Processing
(August to November 2009)
Lecture map for classes 1 to 8
Instructor: Dr. K. Karthik
Lecture 1: Introduction to Digital Signal processing
•
Defining signals as a manifestation of some complex process. Signals cannot exist in isolation and
emerge from complex systems.
Example 1: Speech generation process
Example 2: Production of seismic waves – Arriving seismic waves can be used to characterize the
behaviour of the underlying system (localize the fault and track system behaviour as it traverses
through various states).
Applications (two signal-system models):
(a)
Isolated system model – Earthquake detection, Astronomy.
(b)
Input-Output model – Channel estimation (Noise + ISI problem), Speaker recognition, Locating
and drilling towards oil wells.
Other applications: Signal compression (Fourier series representation).
•
Classification of signals: (a) Continuous vs Discrete, (b) Periodic vs Aperiodic, (c) Deterministic vs
Random. Examples of random signals. Speech utterance if unpredictable can be treated as a random
signal. How does one measure randomness? Presence of noise is an indication of randomness. What is
this property which noise possesses? “Entropy”.
Lecture 2: Derivation of Continuous time Fourier transform (CTFT) for aperiodic
energy signals.
•
Representation of periodic signals. Characterization in terms of Fourier series with convergence
criteria (De-Richlets conditions). Fourier series of a periodic rectangular pulse waveform.
•
What happens to the spectrum as the time period increases towards infinity? --- The discrete spectrum
becomes continuous and the amplitudes of the sinusoids shrink to zero. However the product of the
time period
T
and the Fourier coefficient
k
a
remains finite. This is nothing but the Fourier transform.
Derivation of the Fourier transform starting with the Fourier series of the rectangular pulse.
(
)
∑
∞
−∞
=
∆Ω
→
∆Ω
∆
∞
→
⎥
⎦
⎤
⎢
⎣
⎡
∆Ω
⋅
∆Ω
=
k
t
jk
T
e
k
R
t
r
π
2
lim
)
(
lim
0
(
)
∫
∞
∞
−
Ω
∆
∞
→
Ω
⋅
Ω
=
=
d
e
j
R
t
r
t
r
t
j
a
T
π
2
1
)
(
)
(
lim
•
Definition and properties of the impulse function
)
(
t
δ
. Approximation of the delta function in terms
of the rectangular pulse and the triangular function. Differentiation of the unit step function.
•
Convergence and properties of the CTFT: (a) Transform of impulse/delayed impulse, (b) Convolution,
(c) Modulation, (d) Parseval’s theorem, (e) Sinusoid, (f) Transform domain symmetry properties of a
real signal
)
(
t
x
, (g) Transform of derivative of
)
(
t
x
.
Lecture 3: Sampling theorem and derivation of the Discrete time Fourier transform
from the CTFT
Shannon-Nyquist sampling theorem (
B
F
s
2
>
). Sampling and reconstruction of a bandlimited energy
signal
)
(
t
x
a
.
(
)
)
(
)
(
2
)
(
)
(
ˆ
s
s
k
s
a
a
kT
t
kT
t
B
Sin
kT
x
t
x
−
−
⋅
=
∑
∞
−∞
=
π
π
•
Relating the DTFT to the CTFT. Finding a transform domain representation for the samples
)}
(
{
s
a
nT
x
.