Basics of Signal Processing
Low-pass filter
Allows only frequencies below a specified cutoff point to be passed, above the cut-off frequencies are attenuated
Continuous-time signal
Analog signal
JPEG compression
Applies low pass filtering to chromatic color information. In exchange for maintaining brightness information
Unit Impulse
/delta[n]=1 at 0, and 0 elsewhere
Discrete Cosine Transform
(DCT) Converts each pixel group/matrix into a 2d spectrum
Nyquist frequency
(Nyquist Rate) Discrete-time signal cannot be represent frequencies above one half of this sampling rate.
Region of Convergence
(ROC) region where z-plane converges
Sampling Frequency
(Sampling rate) f_s=1/T
Cascade System
(T_2{T_1{x[n]}}) (x[n]*h_1[n])*h_2[n]=x[n]*(h_1[n]*h_2[n])
Cross-correlation
measures the similarity between two signals or sequences
single precision
32 bit floating point numbers
double precision
64-bit floating-point numbers
Interlaced decomposition
A sample is separated into it's even/odd numbered samples
Discrete-time signal
A sequence of values correlated to an index value n which represents the nth value of that sequence
Parseval's Theorem
A signal must have the same energy in the time domain as in the frequency domain
Impulse Response
A systems response to a unit impulse input, found by applying inverse Fourier Transform
Associative Property
A(x[n]*h[n])=(A*h[n])*x[n]
Offset Binary
Allows for representing numbers away from 0 by forcing a pre-described offset
High-pass filter
Allows only frequencies above a specified cutoff point to be passed, below the cut-off frequencies are attenuated. Maybe converted from low-pass by spectral inversion or spectral reversal
Mean
Average
Moving Average
Averages a number of recent actual values, updated as new values become available
Unsigned integer
Binary system that starts at 0
Digital Signal
Both input and output will be discrete
FFT
Computational version of discrete Fourier Transform, essentially based on interlaced decomposition
Interpolation
Construction of new sampling points between the existing sampling points creating a higher virtual sampling rate
White Noise
Contains equal amount of all possible frequencies; a constant spectral power density
z-transform
Converts the discrete time-domain signal to the complex frequency-domain called the z-plane
Auto-Correlation
Cross-correlation with the signal itself
Unit Step
Cumulative sum of shifted unit impulses
Frequency Response
DTFT of impulse response
Power Spectrum
DTFT of the auto correlation sequence
Sampling Theorm
Describes the connection between continuous-time/analog signal and the discrete time signal.
FIR Filter
Finite Impulse Response filter. Based on convolution between the impulse response and input signal
Two's Compliment
First left to right value is +/- Postive numbers count up from all zeros, negative counts down from all ones
Single Precision (Formula)
First value=S, next 8 are E with a -127 offset, last 23 are M where M=sum of 2^-(on values). And the final value is -1^S*2^E*M
Convolution
Found by reversing the order of one of the inputs and finding the cross-correlation. Result is arranged in reverse order.
Three Ways to Characterize a Filter
Impulse Response, step response, frequency response
Scaled Impulse
Impulse response of memoryless system
IIR Filter
Infinite Impulse Response filter. Uses a recursive filter which have internal feedback from the output signal
Analog Signal
Input is continuous
Stability
Input signal with finite amplitude produces an output signal with finite amplitude. Sum of amplitudes is less than infinity
JPEG
Joint Photographic Experts Group
Band pass filter
May be converted from Band Stop by Spectral Inversion or by cascading low and highpass filters which overlap. Allows medium frequencies to pass
Magnitude Response
Modulus (Magnitude) of the Frequency Response. May be used for defining pass-band, cut-off frequency, transition band and stop band of a low-pass filter, as well as its roll-off sharpness, pass-band ripple and stop-band attenuation in the frequency domain
Fixed-point representation
Negative powers of 10 can be used as a scaling factor to represent non integer values, fixing the decimal point for all numbers
Limit for Analog Signal
Noise and frequency bandwidth
Physically Realizable
Only applies to systems that are both causal and stable
Butterworth Filter
Optimizes passband flatness, sharpest roll-off without passband ripple, has overshoot and ringing
Chebyshev Filter
Optimizes roll-off, has passband ripple, has overshoot and ringing
Bessel Filter
Optimizes step response, slow roll-off, no overshoot or ringing, symmetrical edges, no passband ripple
Delta Modulation
Original analog signal is converted to 1 bit data stream
Causality
Output depends on the present and past input values. All memoryless systems are causal. h[n]=0 for n less than 0. System is stable when all poles are inside the unit circle of the z-plane
Memoryless System
Output depends only on input at that particular moment
Phase Response
Phase angle of the Frequency Response
Median Filter
Replaces the values with in a sampling window with the median value. May function as a low pass filter
Signal to Noise Ration (Fomula)
SNR=10*log_10*(P_signal/P_noise)=20*log_10*(A_signal/A_noise)=10*log_10*(Mean/StD)
Aliasing frequency
Sampling frequency for which the the samples will be indistinguishable for the actual frequency
Limit for Digital Signal
Sampling rate and bit depth
Linear System
Satisfies Additive, and Scaling property
Fourier Transform of Rectangular Pulse
Sinc Function
Sampling period
T=1/f_s
DFT of Scalar Value
The Scalar Value
Central Limit Theorem
The distribution of the sum of non-Gaussian independent random variables is more Gaussian than any of the original variables. I.e. noise follows a normal (Gaussian) distribution
Spectral Reversal
The filter response is reversed
Linearity Property
The magnitude of the output of FFT scales at the same rate as the input
Decimation
The removal of points to reduce that sampling rate
Step Response
The response of a system induced by a unit step input, may be found by taking cumulative sum of impulse response. May be used to describe rise time, overshoot, linear phase in the time domain.
Spectral Inversion
The resulting filter can be thought of as subtracted from the original signal
Band Stop Filter
The sum of a pair of non-overlapping low and high-pass filters, which allows low and high frequencies to pass, but not medium
Digital Filter
Used to separate contaminated signals, or restoring a distorted signal
Standard deviation
Variance
Periodicity Property
X(exp[i(/omega+2Pi)])=X(exp[i(/omega)])
Time/Shift-invariant System
a delay/shift causes and equal sized delay/shift in the response
System
any process that generates an output signal in response to an input signal
Signal
conveys information of how a dependent variable responds to an independent variable
Aliasing frequency (Formula)
f_alias (N)=|N*f_s-f|, where 0<f_alias<f, and N is a positive integer
Transfer Function
fully characterizes the system's properties H(z)=Y(z)/X(z) where the roots numerator represent the zeroes, and the denominator represents poles of the system
correlation coefficient
measures the degree of linear dependency or similarity between two different quantities
Ramp
r[n]=n*u[n] n<0 and 1 for n>0
Signal to Noise Ratio
ration of average signal power to average noise power using logarithmic scale
Unit Step (Formula)
u[n]=0 for n<0 and 1 for n>0
Commutative property
x[n]*h[n]=h[n]*x[n]
Odd Signal
x[n]=-x[-n]
Periodic Sequence
x[n]=n[n+P], where P is the integer that defines the period
Even Signal
x[n]=x[-n]
Compressor System
y[n]=x[Mn], where M is a positive integer
Delay System
y[n]=x[n-n_0]
Running Sum
y[n]=x[n]+y[n-1]
First Difference
y[n]=x[n]-x[n-1]