Edmund Li
PROPERTIES OF THE DTFT
LINEARITY
FREQUENCY SHIFT
TIME SHIFT
Example
[] + [] () + ()
[] ( )
[ 0 ] 0 ()
If a cosine wave of frequency = 6000 is sampled at a rate of = 8000 , sketch the spectrum of
the analogue and discrete signals.
Obviously, ali
Edmund Li
H ALF W AVE D IPOLE A NTENNA
The length of this antenna is and consists of a piece of wire fed at its centre by a generated via a
transmission line. If the antenna length is large compared with the wavelength, then current may not be
considered
Edmund Li
COMPLEX EXPONENTIAL FORM OF DFT
Often, the complex exponential scalar is used to represent:
=
2
This indicates that is one of the N-th root of unity such that = = 1. We thus let be the
vector with elements consisting of the N-th roots of unity
Edmund Li
NYQUIST SAMPLING THEOREM
If the highest frequency contained in an analogue signal () is and the signal is sampled at a rate
2 then () can be exactly recovered from an infinite sequence of samples using an interpolation
function.
2
Quick Proof:
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c)
Now: cos(0 ) = [ 0 0 ]
1
2
1
[cos(0 )] = [ 0 0 ]
2
=
The resulting spectrum is shown:
1
([ 0 ] [ 0 ])
2
= ( 0 ) ( + 0 )
CONVOLUTION
Convolution means folding and is an invaluable tool that provides a mean of finding the response () of a
syste
Edmund Li
DISCRETE TIME FOURIER SERIES (DTFS)
Suppose that we are given a periodic sequence () , then we know that () consists of N
harmonically related exponential fuctions
2
, giving us the DTFS:
1
[] =
=0
With the Fourier coefficients given by:
2
Edmund Li
ANALOGUE TO DIGITAL CONVERSION
Before any DSP algorithm can be performed, a signal must be converted from its analogue form to a digital
form by sampling.
1.
2.
3.
The band limited signal must first be sampled and converted into a discrete time
Edmund Li
PROPERTIES OF THE FOURIER TRANSFORM
LINEARITY
If 1 () and 2 () are the Fourier transforms of 1 () and 2 () respectively:
[1 () + 2 ()] = 1 () + 2 ()
1 [1 () + 2 ()] = 1 () + 2 ()
The same is true for the inverse Fourier transform because it is
Edmund Li
Q UARTER W AVE M ONOPOLE
A quarter wave monopole consists of half of the half wave dipole antenna placed onto a conducting
ground plane. The plane is oriented perpendicular to the antenna and assumed to be infinite. From the
concept of imaging,
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DISCRETE SIGNALS
We describe a discrete signal that has been derived from taking samples of an analogue signal () at discrete
points = as:
[] ()
The constant T referred to as the sampling period whereas the sampling frequency is given by =
= 2
Edmund Li
DIFFERENTIATION IN FREQUENCY
If () = [()] then
() = ()
CONVOLUTION IN TIME
Recall that:
() = () () = ()( )
Then if (), (), () are the Fourier transforms of (), (), () then:
() = [() ()] = ()()
Which indicates that convolution in the time domain