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1
DIGITAL SIGNAL PROCESSING
Lecture 1 - Chapter 1
Classification of Signals: Continuous-Time verses Discrete-Time Signals
Continuous time or analog signals are signals that are defined for every value of a < t <
b, where (a, b) can be ( ∞+∞− , ), i.e., x (t) = e-|t| or ).cos()( ttx π= .
Discrete-time signals are defined at discrete-time instants and between the two discrete
time instants are undefined but are not zero. They can be obtained either by sampling
analog signals or they can be discrete in nature like discrete measurement signals.
A discrete-time signal having a set of discrete values is called a digital signal. Note that
sampling an analog signal produces a discrete-time signal. Then quantization of its
values produces a digital signal.
Deterministic versus Random Signals
Any signal that can be uniquely described by an explicit mathematical expression or a
well-defined rule is called “deterministic”. The past, present and future of a deterministic
signal are known with certainty. Otherwise, it is called “Random” and its properties is
explained by statistical techniques.
Review of Sinusoids in Continuous and Discrete Time
( ) ( )
( ) F2Ft2A
ttAtxa
πΩθπ
θΩ
=+=∞<<∞−+=
cos
cos
( ) ( )F
TtxTtx papa
1, ==+ : fundamental period. Increasing F means increasing
oscillation in time domain. F = 0 corresponds to Tp = ∞ .
2
Also, for complex exponential signals, ( ) ( )θΩ += tja Aetx . A sinusoidal signal then can
also be expressed as
( ) ( ) ( ) ( )θΩθΩθΩ +−+ +=+= tjtja Ae
21
Ae21
tAtx cos
Discrete-Time Sinusoid Signals
( ) ( )( ) ( ) ∞<<∞−+=
=+=nnfAnx
fnAnx
θπ
πωθω
0
0
2cos
2cos
A few important differences between continuous sinusoid and discrete sinusoids:
1) A discrete-time sinusoid is periodic only if its frequency is a rational number.
By default, ( ) ( )nxNnx =+ for all n if x(n) is periodic. The smallest N is called
Fundamental Period.
( ) ( )( ) ( )θπθπ +=++=+ nfNnfANnx 00 2cos2cos
This relationship is true if and only if ππ KNf 22 0 =
NK
f =⇒ 0 : a rational number.
To determine the period N of a periodic discrete time sinusoid, we express f as two
relatively prime numbers. Observe that a small change in frequency can result in a
large change in period. For example, ( ) ( )θπθπ +=
+
= nAnAnx cos
21
2cos1 its
period is 6030
21
1 ===fK
N .
Now consider ( ) 60517.06031
,6031
2cos 222 =→==
+
= NfnAnx θπ
2. An analog ( )+∞∞− ,F maps to 21
f21
≤≤− or equivalently to πωπ ≤≤− or in
other words, the highest rate of oscillation occurs at 21
f += or πω += . To see
what happens for πωπ 2≤≤ consider ω1 = ω0 and ω2 = 2π - ω0 . When ω1 varies
between π to 2π, then ω2 varies between π and 0. Now
( ) nAnAnx 011 coscos ωω ==
3
( ) ( ) ( ) ( )nxnAnAnAnAnx oo 1*
022 coscos2coscos)( ==−=−== ωωωπω
* this is only true because n is an integer value, i.e., x(n) is a discrete signal.
Hence, ( ) ( )nxnx 12 = is an alias because +∞<<∞− F maps only to 21
f21
≤≤−
by sampling.
Analog to Digital Conversion (A/D)
- sampling (sampling rate)
- quantization
Sampling: ( ) ( )
( ) nTsta
sa
tx
nnTxnx
==
∞<<∞−= where,
ss F
nnTt == , Fs = Sampling rate (Frequency) (Hz)
Consider an analog sinusoid: ( ) ( )φπ += Ft2Atxa cos
( ) ( ) ( )
+=+== φπφπ n
FF
2AnTF2AnTxnxs
ssa coscos
Now recall that ∞<<∞− F maps to 21
f21
≤≤−
→21
FF
f21
s≤=≤
− → max2FFs ≥ Nyquist rate
Example 1: ( )( ) Hz50Ft100tx
Hz10Ft20tx
22a
11a
=⇒=
=⇒=
π
π
cos
cos
If we digitize both of these signals with Fs = 40Hz, then
( ) ( ) n2
n40
2040n
xnTxnx 1as1a1ππ
coscos ==
==
( )
( ) !!cos
coscoscos
nxn2
n2
2n2
5n
40100
40n
xnx
1
2a2
==
+===
=
π
ππ
ππ
Therefore, by this sampling rate x2(n) has become same as x1(n), which is an aliasing
error. Equivalently, in this case, the frequency of 50 Hz is an alias of 10 Hz by sampling
max2FFs ≥
4
rate of 40 Hz. Furthermore, all frequencies (F1 + 40K) are aliases of F1. Hence, do not
use the Nyquist rate blindly.
Example 2: ( )43421443442143421
321 xxx
a t100t30010t503tx πππ cossincos −+=
Nyquist rate? f1 = 25, f2 = 150, f3 = 50 Hz
→ Fmax = 150 Hz → Fs = 300 Hz?!
Problem: with Fx = 300 Hz, ( ) 0n10Fn
xnxs
2a2 ==
= πsin all the time!
If it had a phase shift no <<θ , then it would have been fine, but it is best to choose a
higher sampling rate.
Sampling
( ) ( ) ( )tptxtx ap ⋅=
( ) ( )∑+∞
∞−−= snTttp δ
( ) ( ) ( ) ( )
( ) ( ) ( )[ ]
( ) ( )
( ) ( )sas
p
Kss
s
ap
ssnTtp
kXT
X
kT
P
PXX
nTtnTxtxnxx
ωωω
ωωδπ
ω
ωωπ
ω
δ
−=
−=
=
−==
∑
∑
∑
∞+
∞−
∞+
∞−
+∞
∞−=
1
2
*21
|
Therefore, Xp(ω) is a periodic function of shifted the Xa(ω).
x
P(t)
xa(t) x(n)
Ts
x(n)
( )txa
5
( )ωX
Obviously, if Ms ωω 2≥ there is no
aliasing and the signal can be
reconstructed accurately.
In practice however, generating very narrow impulse is very difficult. Therefore, the
practical way for sampling is zero-order hold. Such a system samples xa(t) at a given
sampling instant and holds that value until the succeeding sampling instant.
Reconstruction of xr(t) from the output of this system requires a cascade of low-pass
filters or a non-constant gain of LPF.
• • •
Xp(ω)
ωM ωs
xp(t)It is as if ( )→txa x
P(t) H0(t)( )txo
( )( )ωr
r
H
thxr(t)
|Hr(ω)|
-ωs /2 ωs /2
wωs /2
-ωs /2 2π
<Hr(ω)