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]