summary basics
TRANSCRIPT
-
7/29/2019 Summary Basics
1/12
Brief Notes on 1st Part of ME579
Would like to see X(f), the Fourier Transform of x(t),or
ck, the Fourier Series coefficients, if the time history isperiodic.
We have:x(n) samples of x(t) every seconds forn=0,1,2,3.N-1.
And can do Discrete Fourier Transforms (DFT) to give:Xk for k=0,1,2,.N-1 corresponding to frequenciesfk = 0, fs/N, 2fs/N, .(N-1)fs/N.
-
7/29/2019 Summary Basics
2/12
Brief Notes on 1st Part of ME579
ISSUES: We have done three things
WINDOWEDWINDOWED(TRUNCATION - LEAKAGE)
xw(t) = w(t) . x(t)Xw(f) = W(f) convolved with X(f)
SAMPLED IN TIMESAMPLED IN TIME(ALIASING, CLIPPING, QUANTIZATION NOISE)
xS(t) = (t-n) . x(t)Periodic in Frequency.
-
7/29/2019 Summary Basics
3/12
Brief Notes on 1st Part of ME579
SAMPLED IN FREQUENCYSAMPLED IN FREQUENCY
Xk = XS(f) evaluated at f= fkfk = 0, fs/N, 2fs/N, .(N-1)fs/N.
Signal now becomes Periodic in Time
xn+qN = xn
-
7/29/2019 Summary Basics
4/12
Domains: Time and Frequency
Sampled/Discrete Periodic
Multiplication Convolution
Real and Even Real and Even
Real and Odd Imaginary and Odd
Narrow Broad
Length (T or fs) Resolution (1/T or )
-
7/29/2019 Summary Basics
5/12
When do we Zero Pad Signals?
1. When result of an operation should yield alonger signal than original signal(s).e.g. convolution and time-delay.
2. When we want to have a clearer picture ofXs(f), the Fourier Transform of the sampledsignal, xs(t).
[Would prefer to transform more data to getbetter resolution, i.e., a spikier W(f).Use zero padding when we dont have any moredata to transform.]
-
7/29/2019 Summary Basics
6/12
Discrete Fourier Transform (DFT), XkRelationship to X(f) and ck
Assume no aliasing when sampling [fs > 2 fmax]Have x(n) for n=0,1,2,N-1;
Xk = D.F.T.(x); k=0,1,2,3,N-1.
Periodic Signals:N = a whole number of periods = q Tp
ck = Xk/N for -(N/2) < k < (N/2)
Transients (some aliasing will occur):X(f) X
kfor -fs/2 < f < fs/2
-
7/29/2019 Summary Basics
7/12
Discrete Fourier Transform (DFT), Xk
Symmetric about 0 and fs/2.Real Part:even symmetry about these pointsImaginary Part:odd symmetry about these points
PeriodicXk for k=+(N/2),+(N/2)+1, . N-1
equal toXk for k=-(N/2),-(N/2)+1, . 1
[fftshift will rearrange for you; you have tomake the corresponding frequency vector.]
-
7/29/2019 Summary Basics
8/12
Analog to Digital - Digital to Analog
Input/Output Range, No. of Bits, fs
Analog to Digital Conversion (ADC)
Quantization Error, Clipping, Sample and hold,
Aliasing, Anti-aliasing filters, Sample rate.
f1 = highest frequency of interest
fc = filter cut-off frequencyfmax = highest frequency in filtered signal
fs > 2 fmax = sample rate.
f1 < fc < fmax < fs/2
-
7/29/2019 Summary Basics
9/12
Analog to Digital - Digital to Analog
Digital to Analog Conversion
Similar issues as for ADC
Zero-order hold characteristicssinc function in frequency,
distorts signal in range -fmax> fmax.
Reconstruction Filter: fmax < fc
-
7/29/2019 Summary Basics
10/12
Other Things
We use the delta function,
(t) or
(f),in a lot of our theory and proofs. Sifting property in integrals Integral from - to + of exp(j2ft)
Sampling theory
We looked at the FFT algorithm (not on
exam).
Convolution of continuous and discrete
signals.
-
7/29/2019 Summary Basics
11/12
All the Transforms: timefrequency
Complex Fourier Series x(t) ck
periodic in time, discrete in frequency
Fourier Transforms x(t) X(f)
continuous in time and frequency
Fourier Transform of a sampled signal:
xs(t) Xs(f) OR x(n) Xs(f)
Xs(f) = (1/) q
X(f - q fs)
discrete in time, periodic in frequency
Discrete Fourier Transform (finite set of data used)
xn X
k; n and k: 0,1,2..N-1.
eriodic and discrete in both time and fre uenc
-
7/29/2019 Summary Basics
12/12
Using the Discrete Fourier Transform
Fourier Series coefficients(no aliasing and N corresponds to a whole no. of periods)ck = Xk/N for k=0,1,2N/2.
Approximate the Fourier Transform X(f)of x(t) for frequencies: f = k.f
s
/N(no aliasing)
X(f) at f = k.fs/N = Xk for k=0,1,2(N/2).
Sampled version of Xs(f)(sampled signal, xs(t) was of finite length = N points)
Xs(f) at f = k.fs/N = Xk
Can zero pad to evaluate at more frequency points