ECEn 487 Intro to Digital Signal Processing
General dB drop from main lobe to peak side lobe
-13 dB
Low-pass filter, interpolation
Gain = L, Cutoff = pi/M
Causal ROC
must be exterior to furthest pole (including infinity), extends outwards
Tools for computing Inverse Z-Transform
1. Partial Fraction Expansion 2. Contour Integration 3. Power Series Expansion/Long Division
Methods for Computing Z-transform:
1. Table look up 2. Properties of Z-transforms 3. Closed form Series Representation (when infinitely long series, not in tables)
Periodicity of Fourier Transform
2*pi
Bilinear Transformation
A mapping of the complex plane where one point, s, is mapped into another point, z, through z= (as + b)/(cs + d).
Right-sided sequence
A sequence that is zero for n<N_1<infinity
Left-sided sequence
A sequence that is zero for n>N_2>-infinity
Two-sided Z-transform
Bilateral, Z{x[n]} = summation from n = -infinity to infinity of x[n]*z^(-n)
Overlap Save Method
Convole a finite FIR w/ an infinitely long data, use circular conv y[n] = h[n]*x[n], x_r[n] = x[n + r(L-theta) - theta] 0<=x<=L-1. L = window length R = window index Theta = window overlap, theata = p-1 P = length of h[n] < L h[n] = FIR filter Discard first p-1 samples and then concatenate outputs to get y[n]
DFT
Discrete Fourier Transform, DTFT- closed form that we can represent, exists on unit circle of z-transform, periodic in 2pi, N-point DFT, X[k] = sum from n =0 to N-1 of x[n] * W_n^(k*n).
DTFT
Discrete Time Fourier Transform, z-transform on unit circle
N Place Algorithm
Divide Larger N into smaller stages of easily computed FFT length. (e.g. 100 -> 5*20 -> 5*2*10 -> 5*2*2*5)
Cooley-Tukey Algorithm
Factored N, N = N1*N2. Create an index map, and twiddle factor using N1 and N2
FFT
Fast Fourier Transform, the DFT computed more efficiently
Low-pass Filter, decimation
G = 1, Cutoff = pi/M
Converging Fourier Transform
If ROC included the unit circle
IIR Filter
Includes poles, computational order lower than FIR
Interpolate
Increases f_s. L = 3, inserts L-1 zeros.
Two-sided sequence
Infinite duration sequence
Low pass filter, frequency response
Isolates baseband frequency and creates a copy of it every 2*pi
Decimate
Lowers f_s, drops amplitude to 1/M. M = 3, stretches to M*2*pi, wc = pi/3
All Pass System (based on phase)
Manipulates/controls phase but leaves magnitude alone
Radix 4 Butterfly
Most efficient FFT, because multiplies just 1,-1,j,-j, due to unit circle divided by 4, e^-j*(2*pi/N) -> e^-j*(pi/2)*n
Can the ROC contain any poles?
No
FIR Filter
No poles (not a fraction, no denominator -> no poles), just delays. Simple to design, linear phase
What is needed to go from Z-transform to Time Domain
ROC
Diverging / not converging absolutely Fourier Transform
ROC does not include the unit circle
ROC for left-sided sequence
ROC extends inwards from innermost non-zero pole in X(z) to and maybe including z=0
ROC for right-sided sequence
ROC extends outward from outermost finite pole in X(z) to and maybe including z = infinity
ROC for a finite-duration sequence ( a sequence that is 0 except for a specific finite interval)
ROC is the entire z-plane except possible z=0 or z=infinity
ROC for two-sided sequence
ROC will be a ring bounded on interior and exterior by a pole, but not containing a polee
Poles
Roots of the denominator
Zeros
Roots of the numerator
Gibbs Phenomenon, and in FIR filters
The ringing near a discontinuity in a signal that is caused by incomplete Fourier synthesis, or missing frequencies, in FIR filter design, largest ripple right before transition band.
What happens when a pole and zero are at the same location
They cancel each other out
What's the general relationship between the number of poles and the number of zeros?
They're the same
One-sided Z-Transform
Unilateral, Z{x[n]} = summation from n = 0 to infinity of x[n]*z^(-n)
Rational Sample Rate Change
Upsample (interpolate), then Dowsample (decimate). Gain = L, cutoff = min(pi/L, pi/M),
W_n, twiddle factor
W_n ^(nk)= e^(-j*2pi*n/N)
When can expansion of z-transform to Fourier transform occur?
When z = r*e^(jw)
Rational Function
X(z) = P(z)/Q(z)
Properties of ROC that contains Unit Circle
stable and converges
2 Point DFT
sum and difference, efficient DFT
Fourier Transform
summation from n = -infinity to infinity of x[n]*e^(-jwn) , where w = 2*pi*f
Circular Convolution
z[n] = x[n] Ox y{n} = sum from k =0 to N-1 of y[k]*x[n-k], take x flip it, then multiply it with y aligning zero terms