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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
Δt
n=1 2 3 4 5 6 7 8 .... N
δ(t-7t)
tà
δ(t) h(t)
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
When f=foà G(fo)=0.5à-3dB attenuation 10log(|H(f)|^2)=0dBßà 1.0
10log(|H(f)|^2)=10log (0.5)=-3dBßà 0.5
See Fourier Pair Table (Exponential one-sided)
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
Phase response is not a linear function of frequency at high frequencies
Range of frequencies (<0.5fo) where (almost) no distortion occurs: For example: If fo=10KHz, @ Td(1KHz)=1/2πfo=0.2 msec delay; producing small percentage of phase error.
Different Distortions
Different Distortions (Example) A phase error of 15 degrees for an audio filter at 15KHz would produce a variation (error) in time delay of about 3 micsec:
http://www.wolframalpha.com/input/?i=1%2F(2*pi*15000)*(15*2*pi%2F360)&t=crmtb01
FT for Discrete-Time Signals • For discrete-time signals x [ n ], two alternative frequency
domain representations are extremely useful. – Discrete-time Fourier transform (DTFT)
• Infinite length – infinite sequence of signal – Discrete Fourier transform (DFT)
• Finite length – finite sequence of signal
Discrete Fourier Transform (DFT) • Communication designs usually use computer
simulations based on DFT and IDFT
• Applications of DFT: – uses the DFT to approximate the spectrum continuous W(f) – uses the DFT to evaluate the complex Fourier series coefficients cn
Remember: for each x(n) we generate the equivalent DFT X(n)
The Fast Fourier transform (FFT) • The Fast Fourier transform (FFT) is an extremely efficient
algorithm for computing DFT • The FFT requires that the sequence length N is an integer
power of 2 • To accomplish this we usually append zeros on either side
of discrete-time sequence x [ n ].
Appended Zeros
Application of DFT
Cont. Signal W(t)
Disc. Signal W(n)
Fourier Series CFT DFT
Magnitude Spectrum
(Fourier Coef.) (Cn)=|W(f)|
W(n) @ Δt W(f)
Cn=1/N W(n)
Example: Using DFT to Compute Continuous FT (CFT)
Continuous waveform and it magnitude spectrum
Windowed waveform and its magnitude spectrum – ww(t) is the truncated version of w(t) over [0,T]à we only obtain N (finite) samples
Remember from before:
Example: Using DFT to Compute Continuous FT (CFT)
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
Example: Using DFT to Compute Continuous FT (CFT)
Sampled Windowed waveform and its magnitude spectrum – fs=1/dt
In Summary: • N is the number of sampled points • T is the interval of the interest • Δt is called the time resolution or sample interval = T/N • fs is the sampling frequency (1/Δt ) • B is the highest frequency in w(t) – our waveform • CHECK: fs > 2B • Δf is frequency resolution = 1/T • f represents the frequency points = n/T ; n = [0,1,2, N-1]
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; % sampling period t = n*dt; % simulation time % 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));
Tend = 1 T=10
Pos. Freq. Neg. Freq.
Zoomed to f = [0, 4]
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)
Magnitude Spectrum: |Cn|
10Hz
70Deg. @ 10Hz
fo=10
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)