Feb 23, 2015 ... Advanced, Fifth Edition by Wayne Tomasi – Chapter 2. (https://www.goodreads.
com/book/show/209442.Electronic_Communications_System)
Chapter 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15
Outline • Review of linear systems • Sampling theorem • Fast Fourier Transform
Linear Time Invariant System (LTIS) - 1 L is a Linear Operation
Example: y(t) = t – 3 Is a linear time invariant system
Linear Time Invariant System (LTIS) - 2
h(t)
δ(t)
δ(t-7t) Δt
tà n=1 2 3 4 5 6 7 8 .... N
Linear Time Invariant System (LTIS) - 3
This is called the convolution integral!
Linear Time Invariant System (LTIS) - 4
Example: Linear Time Invariant System (LTIS) - 5
power transfer function (or power gain) of the system
Example: RC Low-Pass Filter Characterization
See Fourier Pair Table (Exponential one-sided)
10log(|H(f)|^2)=0dBßà 1.0
10log(|H(f)|^2)=10log (0.5)=-3dBßà 0.5
When f=foà G(fo)=0.5à-3dB attenuation
Distortionless Transmission -1 • An LTI system is termed distortionless if it introduces the same attenuation to all spectral components and offers linear phase response over the frequency band of interest:
Ho is the gain (or attenuation!) If Ho is unity then there is no lossà Lossless system We refer to to as the Td or time delay
Distortionless Transmission -2
Note that the phase response is a linear function of frequency in LTI! Group delay: refers to time delay that difference spectral components experience!
Distortionless Transmission -3 • The phase delay of an LTI system is defined as
• For a LTI system
(from before)
Is the Output of an RC Filter Distortionless? Remember, for RC filter:
-
Introducing both amplitude and phase distortion! …see next
Is the Output of an RC Filter Distortionless? Amplitude distortion if the amplitude response is not flat
Range of frequencies ( 2B • Δf is frequency resolution = 1/T • f represents the frequency points = n/T ; n = [0,1,2, N-1]
Sampled Windowed waveform and its magnitude spectrum – fs=1/dt
Periodic Sampled Windowed waveform and its magnitude spectrum – fs=1/dt=N/T & dt=T/N (or Period T = N.dt) & fo=1/To
X(n) is the DFT
Using FFT to find the DFT - MATLAB Example M = 7; N = 2^M; % Using zero padding n = 0:1:N-1; T = 10; % period dt = T/N; t = n*dt;
% sampling period % simulation time
Tend = 1
T=10
Zoomed to f = [0, 4]
% Creating time waveform % w=Your waveform! % Calculating FFT W = dt*fft(w); f = n/T; plot(t,w); plot(f,abs(W); plot(f,180/pi*angle(W));
Pos. Freq.
Neg. Freq.
Using DFT to Compute the Fourier Series w
w
Example (MATLAB Implementation)
We use the DFT (FFT) to approximate the spectrum continuous W(f) & evaluate the complex Fourier series coefficients cn
Example (MATLAB Implementation) fo=10
Magnitude Spectrum: |Cn|
10Hz
70Deg. @ 10Hz
Note f=n*fo=10
References • Leon W. Couch II, Digital and Analog Communication Systems, 8th edition, Pearson / Prentice, Chapter 1 • Electronic Communications System: Fundamentals Through Advanced, Fifth Edition by Wayne Tomasi – Chapter 2 (https://www.goodreads.com/book/show/209442.Electronic_Communications_System)