Digital Signal Processing: Complex Numbers, Transforms, and Filters
Complex Numbers
Given
Polar Form
Systems
Sampler
Time
equals
taking samples every T seconds.
Frequency
equals
extensively
and frequency
Decimator
Time
This eliminates one of every M samples.
Frequency
The amplitude scale
and the frequency is multiplied (expanded) by M. The stage is divided by M.
The sampling frequency should be:
Insertaceros
Time
Insert L-1 zeros between samples
Frequency
Divide (compresses) the frequency by a factor L and the amplitude is unchanged. The stage is multiplied by L.
Retardation
Time
Delays the signal
Frequency
The frequency is unchanged, the phase is delayed
.
Relations
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Calculation of the Line Slope
Geometric Series
The way
. Their sum is:
Discrete Fourier Transform
Definition
Reverse
FFT Using Decimation in Time
Samples are taken of the signal pairs
and sent to a DFT
points, and does the same with the odd samples. The output of the DFT with the odd samples is multiplied by
and added to the sample pairs. In turn, the DFT
points can be solved in the same way, by recursively until a two-point DFT, resulting in:
Z Transform
- Zeros: roots of the numerator
- Polos: roots of the denominator
Definition
? TF Value TZ
Inverse Z Transform
Given
.
If the degree of the numerator is greater than or equal to the denominator:
It performs the polynomial division
to be of the form
The result will be:
If not:
and decomposes into simple fractions of the form
where
Is recalculated
- Calculate the inverse Z transform of each term:
Information Systems
Property | Freq Response | Impulse response | Diagram of zeros and poles |
Stable | The resp. is summable | The ROC is inside the circle of unit radius. | |
Causal | Zero for negative values | The ROC is external to the last pole. | |
Real | Real coefficients | The poles are conjugate | |
Step all | The resp. is constant | The poles are zeros in their inverse conjugate | |
FIR | The resp. is finite | The poles are at 0 or
| |
IIR | The resp. is infinite | ||
Minimum Phase | All poles and zeros are inside the unit radius circle (not the border). | ||
Linear phase | It is symmetric FIR | All poles have zero in its inverse | |
Realizable (stable and causal) | All poles are inside the unit radius circle. |
Digital Filter Design
Represent the attenuation ? (dB) and logarithmic scale
(Gain, G =- A in db, in linear g = 1 / a), as well as analog specifications | H (?) | (after having computed ? p and ? a).
Get Specs analog (
). We can do this using two methods:
- Impulse Invariance:
,
for convenience
- Bilinear transformation:
for convenience
The alphas have the same value, you do not make any transformation.
- We calculate the order (N):
The review N = 2, if not, review the accounts.
- We calculate the frequency:
- To design the analog filter Butterworth asked to use us:
Only use stable k s are in the left half. We obtain the s k and substitute into the equation H (s). Calculate until H (s) only depends on s and other variables are numerical.
- Undo the transformation frequency:
- Impulse Invariance Method:
If
:
- If not:
Reduced H (s) to simple fractions of the form
- Calculate:
- Bilinear Transformation Method:
We check to calculate H (z) whether or not the filter standard, ie if the number is no z in the denominator is greater than one, and if so divide everything between this value to normalize the filter.
- Once we have H (z), we make the block diagram in canonical form, with the difference equation (the difference equation is for the time domain).
The degree of z is the number of delay