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 is periodic. We have: x(n ) samples of x(t) every seconds for n=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 frequencies fk = 0, fs/N, 2fs/N, .(N-1)fs/N.
Brief Notes on 1st Part of ME579
ISSUES: We have done three things WINDOWED
(TRUNCATION - LEAKAGE)
xw(t) = w(t) . x(t) Xw(f) = W(f) convolved with X(f) SAMPLED IN TIME
(ALIASING, CLIPPING, QUANTIZATION NOISE)
Periodic in Frequency.
xS(t) = (t-n) . x(t)
Brief Notes on 1st Part of ME579
SAMPLED IN FREQUENCY
Xk = XS(f) evaluated at f= fk
fk = 0, fs/N, 2fs/N, .(N-1)fs/N.
Signal now becomes Periodic in Time
xn+qN = xn
Domains: Time and Frequency
Sampled/Discrete Multiplication Real and Even Real and Odd Narrow Length (T or fs) Periodic Convolution Real and Even Imaginary and Odd Broad Resolution (1/T or )
When do we Zero Pad Signals?
1. When result of an operation should yield a longer signal than original signal(s). e.g. convolution and time-delay. 2. When we want to have a clearer picture of Xs(f), the Fourier Transform of the sampled signal, xs(t). [Would prefer to transform more data to get better resolution, i.e., a spikier W(f). Use zero padding when we dont have any more data to transform.]
Discrete Fourier Transform (DFT), Xk Relationship to X(f) and ck
Assume no aliasing when sampling [fs > 2 f max]
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) Xk for -fs/2 < f < fs/2
Discrete Fourier Transform (DFT), Xk
Symmetric about 0 and fs/2.
Real Part: even symmetry about these points Imaginary Part: odd symmetry about these points
Periodic
Xk for k=+(N/2),+(N/2)+1, . N-1 equal to Xk for k=-(N/2),-(N/2)+1, . 1 [fftshift will rearrange for you; you have to make the corresponding frequency vector.]
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 frequency fmax = highest frequency in filtered signal fs > 2 fmax = sample rate.
f1 < fc < fmax < fs/2
Analog to Digital - Digital to Analog
Digital to Analog Conversion Similar issues as for ADC Zero-order hold characteristics sinc function in frequency, distorts signal in range -fmax<f<fmax, distortion less as fs >>> fmax. Reconstruction Filter: fmax < fc << fs/2.
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.
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) discrete in time, periodic in frequency Discrete Fourier Transform (finite set of data used) xn Xk; n and k: 0,1,2..N-1. periodic and discrete in both time and frequency Xs(f) = (1/ )
X(f - q fs)
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.fs/N
(no aliasing)
X(f)
at f = k.fs/N
= Xk
for k=0,1,2(N/2).
Sampled version of X s(f) Xs(f)
(sampled signal, xs(t) was of finite length = N points)
at f = k.fs/N
= Xk
Can zero pad to evaluate at more frequency points
