Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT...

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Chapter 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15

Transcript of Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT...

Page 1: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

Chapter 2 – Part 2

Liner System Review, DFT & FFT Updated:2/23/15

Page 2: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

Outline •  Review of linear systems •  Sampling theorem •  Fast Fourier Transform

Page 3: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

Linear Time Invariant System (LTIS) - 1

L is a Linear Operation

Example: y(t) = t – 3 Is a linear time invariant system

Page 4: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

Linear Time Invariant System (LTIS) - 2

Δt

n=1 2 3 4 5 6 7 8 .... N

δ(t-7t)

δ(t) h(t)

Page 5: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

Linear Time Invariant System (LTIS) - 3

This is called the convolution integral!

Page 6: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

Linear Time Invariant System (LTIS) - 4

Page 7: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

Example: Linear Time Invariant System (LTIS) - 5

power transfer function (or power gain) of the system

Page 8: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

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)

Page 9: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

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

Page 10: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

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!

Page 11: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

Distortionless Transmission -3 •  The phase delay of an LTI system is defined as

•  For a LTI system

(from before)

Page 12: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

Is the Output of an RC Filter Distortionless? Remember, for RC filter:

-

Introducing both amplitude and phase distortion! …see next

Page 13: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

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.

Page 14: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

Different Distortions

Page 15: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

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

Page 16: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

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

Page 17: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

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)

Page 18: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

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

Page 19: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

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)

Page 20: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

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:

Page 21: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

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

Page 22: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

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

Page 23: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

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]

Page 24: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

Using DFT to Compute the Fourier Series

w w

Page 25: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

Example (MATLAB Implementation)

We use the DFT (FFT) to approximate the spectrum continuous W(f) & evaluate the complex Fourier series coefficients cn

Page 26: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

Example (MATLAB Implementation)

Magnitude Spectrum: |Cn|

10Hz

70Deg. @ 10Hz

fo=10

Note f=n*fo=10

Page 27: Chapter 2 – Part 2 - Sonoma State University 2 – Part 2 Liner System Review, DFT & FFT Updated:2/23/15 . Outline • Review of linear systems ... Advanced, Fifth Edition by Wayne

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)