1 Sampling Theorem © Ammar Abu-Hudrouss Islamic University Gaza
1
Sampling Theorem
© Ammar Abu-Hudrouss Islamic
University Gaza
Slide 2
Digital Signal Processing
Signal is any physical quantity that varies with time, space, or any other variable.
System is a physical device or software that performs an operation on a signal
Slide 3
Digital Signal Processing
Basic Elements of DSP System
Analogue electronic systems are continuous
Electronic System are increasingly digitalized
Signals are converted to numbers, processed, and converted back
Analogue System x(t) y(t)
Digital System A/D D/A y(t) x(t) y(n) x(n)
Slide 4
Digital Signal Processing
Advantages of Digital over Analogue
Advantages
Flexibility (simply changing program).
Accuracy.
Storage Capability.
Cheap
Ability to apply highly sophisticated algorithms.
Disadvantages
It has certain limitations (very fast sample rate is needed when the bandwidth of signal is very large)
It has a larger time delay compared to the analogue.
Slide 5
Digital Signal Processing
Classification of signals
Mono-channel versus Multi-channel
One Dimensional versus Multidimensional
Continues Time versus Discrete Time
Continuous Values and Discrete Valued
Deterministic versus Random
Slide 6
Digital Signal Processing
Periodic Continuous Signal
21
FT
tAtx cos)(
We will take sinusoidal signals for example. Continuous sinusoidal signal has the form
The signal can be characterised by three parameters A: Amplitude, frequency in radian and : phase
The period is defined as,
Slide 7
Digital Signal Processing
Periodic Continuous Signal
7
In analogue signal, increasing the frequency will always lead to increase the rate of the oscillation.
In analogue signals with distinct frequencies are themselves distinct from each other.
Slide 8
Digital Signal Processing
Periodic Discrete Signal
)22cos()2cos(
)()(
fNfnfn
Nnxnx
nAnx cos)(
N
kf
kkfN
,......2,1,022
8
Discrete sinusoidal signal has the form
1) Discrete time sinusoid is periodic only if its frequency in hertz ( f = / 2) is a rational number
From the definition of a periodic discrete signal
This is only true if
Slide 9
Digital Signal Processing
Periodic Discrete Signal
)()cos())2cos((1 nxnAnAnx
nAnx cos)(
,......2,1,02 kkk )cos()( nAnx kk
9
2) Discrete time sinusoid whose radian frequencies are separated by integer multiples of 2 are identical
To prove this, we start from the signal
As a result, all the following signals are identical
3) All signal in the range - < are unique.
So the range of the discrete frequency f is [-0.5 0.5]
Slide 10
Digital Signal Processing 10
Slide 11
Digital Signal Processing
Harmonically related Complex Exponential
The basic signals for continuous-time harmonically related exponentials are
tjk
k ets 0
Nfk /1,2,1,0 0
The basic signals for discrete-time harmonically related exponentials are
nkfj
k ens 02
00 2,2,1,0 Fk
Slide 12
Digital Signal Processing
Analogue to Digital Conversion
Sampler Quantizer Coder xa(t) x(n) xq(n)
Analog Signal
Discrete-time Signal
Quantized Signal
Digital Signal
101101…
1) Sampling: Conversion of analogue signal into a discrete signal by taking sample at every Ts s.
2) Quantization: Conversion of discrete signal into discrete signals with discrete values. (the value of each sample is represented by a value selected from a finite set of possible value)
3) Coding: is process of assigning each quantization level a unique binary code of b bits.
Slide 13
Digital Signal Processing
Sampling of Analog Signal
We will focus on uniform sampling where
x (n) = xa(nTs) -∞ < n < ∞
Fs = 1/Ts is the sampling rate given in samples per second
As we can see the discrete signal is achieved by replacing the continuous variable t by nTs.
Consider the analog signal xa (t ) = A cos(2Ft + ) The sampled signal is xa(nT) = A cos(2FnTs + ) x (n) = A cos(2fn + ) The digital frequency = analog freq. X sampling time
f = FTs
Slide 14
Digital Signal Processing
Sampling of Analog Signal
But from previous discussion , for the analogue frequency
-∞< F <∞ or -∞< <∞
And for the digital frequency
-0.5 ≤ f < 0.5 or - ≤ <
From the above argument the infinite analog frequency is mapped into finite digital frequency.
This mapping is one-to-on as long as the resultant digital frequency is between the limits of [-0.5 o.5]
Slide 15
Digital Signal Processing
Sampling of Analog Signal
Which leads that
-1/2 ≤ FTs < 1/2 or - ≤ Ts <
OR
-1/(2Ts) ≤ F < 1/(2Ts) or - /Ts ≤ < /Ts
Hence that highest possible analogue frequency is
Fmax = Fs /2 = 1/(2Ts) and < Fs = /Ts .
Fs /2 is called the folding frequency
Slide 16
Digital Signal Processing
Sampling of Analog Signal
Example Consider the two analog sinusoidal signals
X1(t ) = cos [2(10)t ] and X2(t ) = cos [2(50)t ]. Both are sampled with sampling rate Fs = 40 sample/s, find the
corresponding discrete sequences
X1 (n) = cos [2(10/40)t] = cos [n/2]
X2(t) = cos [2(50/40)t] = cos [5n/2] = cos [n/2]
a 1Hz and a 6Hz sinewave are sampled at a rate of 5Hz.
Slide 17
Digital Signal Processing
Sampling of Analog Signal
All sinusoids with frequency
Fk = F0 + k Fs, k = 1,2,3,………
Leads to unique signal if sampled at Fs sample/s.
proof
xa(t) = cos (2 Fk t + ) = cos (2 (F0 + k Fs )t +)
x(n) = xa(nTs) = cos (2 (F0 + k Fs )/Fs n + ) = cos (2 F0 /Fs n + 2 k n + ) = cos (2 F0 /Fs n + ) = cos (2 f0 n + )
Slide 18
Digital Signal Processing
Sampling Theorem
Sampling Theorem
A continuous-time signal x(t) with frequencies no higher than fmax (Hz) can be reconstructed EXACTLY from its samples x[n] = x(nTs), if the samples are taken at a rate fs = 1/Ts that is greater than 2 fmax.
Consider a band-limited signal x(t) with Fourier Transform X()
Slide 19
Digital Signal Processing
Sampling Theorem
Sampling x(t) is equivalent to multiply it by train of impulses
X
Slide 20
Digital Signal Processing
Sampling Theorem
In mathematical terms
Converting into Fourier transform
)()()( tstxtxs
n
ss nTttxtx )()()(
n
s
s
s nT
XX )(1
*)(
n
s
s
s nXT
X )(1
)(
Slide 21
Digital Signal Processing
Sampling Theorem
By graphical representation in the frequency domain
X
Slide 22
Digital Signal Processing
Sampling Theorem
Therefore, to reconstruct the original signal x(t), we can use an ideal lowpass filter on the sampled spectrum
This is only possible if the shaded parts do not overlap. This means that fs must be more than TWICE that of B.
Slide 23
Digital Signal Processing
Sampling Theorem
Example x(t ) and its Fourier representation is shown in the Figure.
If we sample x(t) at Fs = 20,10,5
1) Fs = 20 x (t ) can be easily
recovered by LPF
Slide 24
Digital Signal Processing
Sampling Theorem
2) Fs = 10 x(t ) can be
recovered by sharp LPF
3) Fs = 5 x(t) can not be
recovered
Compare fs with 2 B in each case
Slide 25
Digital Signal Processing
Anti-aliasing Filter
To avoid corruption of signal after sampling, one must ensure that the signal being sampled at fs is band-limited to a frequency B, where B < fs/2.
Consider this signal spectrum:
After sampling:
After reconstruction:
Slide 26
Digital Signal Processing
Anti-aliasing Filter
Apply a lowpass filter before sampling:
Now reconstruction can be done without distortion or corruption to lower frequencies:
Sampler Anti-aliasing
filter x(t)
y(n) x'(t)
Slide 27
Digital Signal Processing
Homework
Students are encouraged to solve the following questions from the main textbook
1.2, 1.3, 1.4, 1.5, 1.7, and 1.9