ECE 308 -2
ECE 308
Continuous-Time and Discrete-Time Signal
Sampling of Analog Signals
Z. Aliyazicioglu
Electrical and Computer Engineering Department
Cal Poly Pomona
ECE 308-2 1
Continuous Time Signal
Let’s have the following continuous-time sinusoidal signal:
xa (t ) = A cos(Ωt + θ ), − ∞ < t < ∞
where
A: the amplitude of the signal
Ω: the frequency in radians per second
θ: the phase in radians
The frequency can be expressed in cycles/s or Hertz (Hz)
F=
Ω
2π
The period is define as
Tp =
1
F
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1
Continuous Time Signal
Tp
A
θ
The analog sinusoidal signal can repeat every period
xa (t + Tp ) = xa (t )
• Increasing the frequency means decreasing the period of the
signal, so that increase the rate of oscillation of the signal
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Continuous Time Signal
The analog sinusoidal signal can be expressed in complex
exponent for as
xa (t ) = A cos(Ωt + θ ) =
A j ( Ωt +θ ) A − j ( Ωt +θ )
e
+ e
2
2
Im
Ω
A/2
Ωt+θ
Re
Ωt+θ
A/2
Ω
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Discrete-Time Sinusoidal Signal
A discrete-time sinusoidal signal may be expressed as
x(n) = A cos(ω n + θ ), − ∞ < n < ∞
where n : integer variable
A : the amplitude of the signal
ω : the frequency in radians per sample
θ : the phase in radians
The frequency can be expressed in cycles per sample
f =
ω
2π
and the signal is
x(n) = A cos(2π fn + θ ), − ∞ < n < ∞
ECE 308-2 5
Discrete-Time Sinusoidal Signal
Example:
A sinusoidal signal with the amplitude A, frequency
ω=π/6 radians per sample (f=1/12) and phase θ=π/3
x(n)
……
n
A discrete-time sinusoidal is periodic only if its frequency is
rational number
x(n + N ) = x( n) cos[2π f 0 ( N + n) + θ ] = cos[2π f 0 n + θ ]
It is true if and only if
2π f 0 N = 2kπ
or
f0 = k / N
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3
Discrete-Time Sinusoidal Signal
Discrete-time sinusoids whose frequencies are separated by an
integer multiple of 2π are identical
cos[(ω 0 + 2π ) n + θ ] = cos(ω 0 n + 2π n + θ ) = cos(ω 0 n + θ )
xk (n ) = A cos(ω k n + θ ) = A cos[(ω0 + 2kπ )n + θ ], for k = 0,1,2...
are identical and where −π ≤ ω0 ≤ π
The highest rate of oscillation in a discrete-time sinusoidal
is attained when ω=π (or ω=-π) equivalent to f=1/2 (or f=-1/2)
x(nFor
) = A cos ω 0 n
ω
0
π/8
π/4
π/2
π
f
0
1/16
1/8
1/4
1/2
N
∞
16
8
4
2
ECE 308-2 7
Discrete-Time Sinusoidal Signal
If π ≤ ω0 ≤ 2π , it creates an aliasing. How?
Let’s ω1 = ω 0 which π ≤ ω 0 ≤ 2π and ω 2 = 2π − ω 0 which π ≤ ω0 ≤ 2π
x1 (n) = A cos ω1n = A cos ω 0 n x2 (n) = A cos ω 2 n = A cos(2π − ω 0 )n
= A cos ω 0 n
= x1 (n)
Hence, ω 2 is an alias of ω1 .
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Analog-to-Digital Conversion (ADC)
• In many real-world application, the signals are analog.
• To process analog signal by digital, we need to convert them
into digital signal
• This process is called Analog-to-Digital conversion and
devices are A/D Converter (ADCs).
• A/D Conversion has three steps:
xa(t)
Sampler
Analog
Signal
Sampling :
x(n)
Quantizer
Discrete-Time
Signal
xq(n)
Coder
Quantized
Signal
Digital
Signal
• Conversion of a continuous-time signal into a
discrete-time signal
• Taking “samples” of the continuous-time signal at
discrete-time instants.
x(n) = xa ( nT )
• Sampling interval is T.
ECE 308-2 9
Analog-to-Digital Conversion (ADC)
2. Quantization:
•
Conversion of a discrete-time continuous valued signal into a
discrete-time, discrete valued digital signal xq (n)
Digital signal values are a finite set of possible values.
The differences between xq ( n) and x( n) ( xq (n) − x(n) ) is called
the quantization error.
Discrete–time Signals are defined only at certain specific values of
time or variable.
•
•
•
3. Coding:
•
Each discrete value is represented by a b-bit binary sequence.
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Sampling of Analog Signals
Analog
Signal
xa(t)
x(n)=xa(nT)
Discrete-time
Signal
Fs=1/T
>> x=0:0.1:10;
>> y=x.^3-18*x.^2+81*x;
>> plot(x,y)
>> x=0:10;
>> y=x.^3-18*x.^2+81*x;
>> stem(x,y)
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Sampling of Analog Signals
The discrete-time signal x(n) is obtained by “taking-samples” of the
analog signal xa (t ) every T second.
x ( n) = xa ( nT )
The time interval T is called the sampling period or sampling interval
The sampling rate or the sampling frequency is found as
Fs =
1
[ Hz ]
T
The relationship between the variable t of analog signal and the
variable n of discrete-time signal is
t = nT =
n
Fs
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Sampling of Analog Signals
Consider an analog sinusoidal signal
xa (t ) = A cos(2π Ft + θ )
Sampling frequency is Fs = 1/ T , so that
x( n) = xa (nT ) = A cos(2π FnT + θ )
= A cos(2π F
n
+θ )
Fs
x(n) = A cos(ω n + θ )
or
f =
We call relative or normalize frequency that
Equivalently,
ω=
F
Fs
2π F Ω
=
= ΩT
Fs
Fs
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Sampling of Analog Signals
Relations between analog signals and Discrete-time signal
Continuous-time signal
Discrete-time signal
ω = 2π f
ω [radians/sample]
Ω = 2π F
f [cycles/sample]
Ω [radians/s]
F [Hz]
Range
−∞ < Ω < ∞
ω =ΩT , f = F / Fs
−π ≤ ω ≤ π
1
2
− ≤f≤
Ω =ω /T , F = f .Fs
1
2
Range
−
−∞ < F < ∞
−
π
T
≤Ω≤
π
T
Fs
F
≤F≤ 2
2
2
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Sampling of Analog Signals
The fundamental different between analog signal and discrete-time
signal is frequency range.
The highest frequency in the discrete-time signal is ω = π or , f = 1/ 2
the sampling rate Fs , the corresponding highest value of F and
are
Fmax =
Example:
Fs
1
=
2 2T
Ω max = π Fs =
Ω
π
T
Consider two analog signal
x1 (t ) = cos 2π 10t
x2 (t ) = cos 2π 50t
The sampling rate is Fs = 40Hz
Find discrete time signal x1(n) and x2(n)
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Sampling of Analog Signals
Example:(cont)
Corresponding discrete-time signals are
π
 10 
x (n) = cos 2π   n = cos n
40
2
 
5π
 50 
x(n) = cos 2π   n = cos n
2
 40 
We know that
cos
Hence
5π
π
π

n = cos  2π +  n = cos n
2
2
2


x1 (n) = x2 (n)
The frequency F2 = 50Hz is an alias of the frequency F1 = 10Hz
at the sampling rate of Fs = 40 Hz. Even
Fk = ( F1 + 40 k ), k = 1,2,3,...
are alias of F1 at the sampling rate at Fs = 40 Hz.
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Sampling of Analog Signals
In general form, Fk = ( F0 + kFs ), k = ±1, ±2, ±3,... are creates an alias for
frequency F0 of analog signal, which are outside of
−
Fs
F
≤F≤ 2
2
2
ECE 308-2 17
Sampling of Analog Signals
F=10 Hz
F=50 Hz
T=0.025 s
>> t=0:0.001:0.2;
>> x1=cos(2*pi*10*t);
>> plot (t,x1,'--')
>> hold on
>> x2=cos(2*pi*50*t);
>> plot (t,x2,'r--')
>> n=0:0.025:0.2;
>> y1=cos(2*pi*10*n);
>> stem (t,y1,'g-')
>> Title('Discrete-time signal with x1(t) and x2(t)')
Fs=40 Hz
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