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