The module provides a review of the background concepts needed for the study of analog and digital communications: the Fourier transform, various definitions of bandwith, the Dirac delta, frequency-domain representation of sinusoids, frequency-domain plotting in Matlab, linear time invariant (LTI) systems, linear filtering, lowpass filters, and Matlab design of lowpass filters.
Fourier transform (ft)
Definition:
Properties:
Linearity:
.
Real-valued
.
“bandwidth”
Dirac delta (or “continuous impulse”)
An infinitely tall and thin waveform
with unit area :
that's often used to “kick” a system and see how it responds.
Key properties:
Sifting:
.
Time-domain impulse
has a flat spectrum:
Freq-domain impulse
corresponds to a DC waveform:
Frequency-domain representation of sinusoids
Notice from the sifting property that
Thus, Euler's equations
and the Fourier transform pair
imply that
Often we draw this as
Frequency domain via matlab
Fourier transform requires evaluation of an integral.
What do we do if we can't define/solve the integral?
Generate (rate-
) sampled signal in MATLAB.
Plot magnitude of Discrete Fourier Transform (DFT)
using
plottf.m (from course webpage).
Notice that
plottf.m only plots frequencies
.
Linear time-invariant (lti) systems
An LTI system can be described by either its “impulse response”
or its “frequency response”
.
Input/output relationships:
Time-domain: Convolution with impulse response
Freq-domain: Multiplication with freq response
Linear filtering
Freq-domain illustration of LPF, BPF, and HPF:
Lowpass filters
Ideal non-causal LPF (using
):
Ideal LPF with group-delay
t
o :
A causal linear-phase LPF with group-delay
t
o :
but MATLAB can give better causal linear-phase LPFs...
In MATLAB, generate
-sampled LPF impulse response via
h = firls(Lf, [0,fp,fs,1], [G,G,0,0])/Ts;
where...
The commands
firpm and
fir2 have the same interface,
but yield slightly different results (often worse for our apps).
In MATLAB, perform filtering on
-sampled signal
x via
y = Ts*filter(h,1,x); or
y = Ts*conv(h,x);
The routines
firls,firpm,fir2 generate
causal linear-phase filters with group delay
samples.
Thus, the filtered output
y will be delayed by
samples relative to
x .
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Source:
OpenStax, Introduction to analog and digital communications. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10968/1.2
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