Sampling and Discrete Time
Sampling is the acquisition of the values of a continuous-time signal
()
at discrete points in time. x t is a continuous-time signal, x ⎡⎣ n ⎤⎦ is a
discrete-time signal.
Discrete-Time Signal
Description
( )
x ⎡⎣ n ⎤⎦ = x nTs where Ts is the time between samples
M. J. Roberts - All Rights Reserved.
Edited by Dr. Robert Akl
1
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
2
Sinusoids
Sampling and Discrete Time
The general form of a periodic sinusoid with fundamental
period N 0 is g [ n ] = A cos ( 2π F0 n + θ ) or A cos ( Ω0 n + θ )
Unlike a continuous-time sinusoid, a discrete-time sinusoid is
not necessarily periodic. If a sinusoid has the form
g [ n ] = A cos ( 2π F0 n + θ ) then F0 must be a ratio of integers
(a rational number) for g [ n ] to be periodic. If F0 is rational in
the form q / N 0 and all common factors in q and N 0
have been cancelled, then the fundamental period of the sinusoid is
N 0 , not N 0 / q (unless q = 1).
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
3
Sinusoids
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
4
Sinusoids
An Aperiodic Sinusoid
Periodic
Periodic
Periodic
Aperiodic
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
5
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
6
1
Sinusoids
Sinusoids
Two sinusoids whose analytical expressions look different,
g1 [ n ] = A cos ( 2π F01n + θ ) and g 2 [ n ] = A cos ( 2π F02 n + θ )
may actually be the same. If
F02 = F01 + m, where m is an integer
then (because n is discrete time and therefore an integer),
A cos ( 2π F01n + θ ) = A cos ( 2π F02 n + θ )
(Example on next slide)
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
7
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
Sinusoids
Exponentials
Complex α
Real α
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
8
9
The Unit Impulse Function
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
10
The Unit Sequence Function
⎧1 , n = 0
δ [n] = ⎨
⎩0 , n ≠ 0
⎧1 , n ≥ 0
u[n] = ⎨
⎩0 , n < 0
The discrete-time unit impulse (also known as the “Kronecker
delta function”) is a function in the ordinary sense (in contrast
with the continuous-time unit impulse). It has a sampling property,
∞
∑ Aδ [ n − n ] x [ n ] = A x [ n ]
0
0
n=−∞
but no scaling property. That is,
δ [ n ] = δ [ an ] for any non-zero, finite integer a.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
11
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
12
2
The Signum Function
The Unit Ramp Function
⎧1 , n > 0
⎪
sgn ⎡⎣ n ⎤⎦ = ⎨0 , n = 0
⎪−1 , n < 0
⎩
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
n
⎧n , n ≥ 0 ⎫
ramp [ n ] = ⎨
⎬ = n u [ n ] = ∑ u [ m − 1]
m=−∞
⎩0 , n < 0 ⎭
13
14
Scaling and Shifting Functions
The Periodic Impulse Function
δ N [n] =
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
Let g [ n ] be graphically defined by
∞
∑ δ [ n − mN ]
m=−∞
g[n ] = 0 , n > 15
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
15
Scaling and Shifting Functions
Time shifting
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
16
Scaling and Shifting Functions
n → n + n0 , n0 an integer
Time compression
n → Kn
K an integer > 1
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
17
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
18
3
Scaling and Shifting Functions
Scaling and Shifting Functions
There is a form of time expansion that is useful. Let
Time expansion
⎧x [ n / m ] , n / m an integer
y[n] = ⎨
, otherwise
⎩0
All values of y are defined.
n → n / K, K > 1
This type of time expansion
is actually used in some
digital signal processing
For all n such that n / K is an integer, g [ n / K ] is defined.
For all n such that n / K is not an integer, g [ n / K ] is not defined.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
operations.
19
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
Differencing
20
Accumulation
g[ n ] =
n
∑ h[m]
m=−∞
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
21
ge [n] =
g [ n ] + g [ −n ]
2
g [ n ] = − g [ −n ]
go [n] =
22
Products of Even and Odd
Functions
Even and Odd Signals
g [ n ] = g [ −n ]
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
Two Even Functions
g [ n ] − g [ −n ]
2
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
23
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
24
4
Products of Even and Odd
Functions
Products of Even and Odd
Functions
An Even Function and an Odd Function
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
Two Odd Functions
25
Symmetric Finite Summation
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
26
Periodic Functions
A periodic function is one that is invariant to the
change of variable n → n + mN where N is a period of the
function and m is any integer.
The minimum positive integer value of N for which
g [ n ] = g [ n + N ] is called the fundamental period N 0 .
N
N
n=− N
n=1
∑ g [ n ] = g [ 0 ] + 2∑ g [ n ]
N
∑ g[n] = 0
n=− N
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
27
Periodic Functions
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
28
Signal Energy and Power
The signal energy of a signal x [ n ] is
Ex =
∞
∑ x[n]
2
n=−∞
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
29
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
30
5
Signal Energy and Power
Signal Energy and Power
Find the signal energy of
x [ n ] = 0 , n > 31
2n
⎪⎧( 5 / 3) , 0 ≤ n < 8
x[n] = ⎨
, otherwise
⎩⎪0
Ex =
∞
∑ x[n]
2
n = −∞
7
7
2n
4
= ∑ ⎡⎣( 5 / 3) ⎤⎦ = ∑ ⎡⎣( 5 / 3) ⎤⎦
2
n=0
n
n=0
, r =1
⎧N
⎪
Using ∑ r n = ⎨1 − r N
, r ≠1
n=0
⎩⎪ 1 − r
N −1
Ex =
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
31
Signal Energy and Power
4
1 − ⎡⎣( 5 / 3) ⎤⎦
1 − ( 5 / 3)
4
8
≅ 1.871 × 10 6
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
32
Signal Energy and Power
Some signals have infinite signal energy. In that case
It is usually more convenient to deal with average signal
power. The average signal power of a signal x [ n ] is
Px = lim
N →∞
2
1 N −1
∑ x[n]
2N n=− N
For a periodic signal x [ n ] the average signal power is
Px =
2
1
∑ x[n]
N n= N
⎛ The notation ∑
means the sum over any set of ⎞
n= N
⎜
⎟
⎝ consecutive n 's exactly N in length.
⎠
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
33
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
34
Signal Energy and Power
A signal with finite signal energy is
called an energy signal.
A signal with infinite signal energy and
finite average signal power is called a
power signal.
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
35
6