Signals: Spectra & Processing Flashcards
The most important piece that needs to be shared and safeguarded from noise.
Signal
An action, a gesture, or sign used as a means of communcation
Means of Communication
A piece of information communicated by an action, gesture, or sign
Communicated Infromation
Something that incites somebody to action
Incitement
Information transmitted by means of a modulated current or an electromagnetic wave and received by telephone, telegraph, etc.
Electronics Transmitted Infromation
It is defined as ta single-valued function of time that conveys information
Signal
Describes the frequency content of the signal
Spectra
Condition that is not limited to a specific set of values but can vary infinitely within a continuum
Spectrum
Separation of visible light through a prism forming a rainbow
Prismatic Diffraction
Any variable signal continuous in both time and amplitude
Analog Signal
Electrical properties used to a signal
Voltage
Frequency
Current
Charge
The digital representation of discrete-time signal which is often derived from analog signal
Digital Signal
A sampled version of an analog signal
Discrete-time Signal
The result of individual time values of the discrete-time signal being approximated to a certain precision
Digital Signal
The process of approximating the precise value within a fixed number of digits or bits
Quantization
It is quantized discrete-time signal
Digital Signal
Any signal processing conducted on analog signals by analog means
Analog Signal Processing
The study of signals in a digital representation and the processing methods of these signals
Digital Signal Processing
The four major subfields of DSP
Audio Signal Processing
Control Engineering
Digital Image Processing
Speech Processing
The domains of digital signals
Time Domain Spatial Domain Frequency Domain Autocorrelation Domain Wavelet Domain
It can be produced by a sequence of samples from a measuring device
Time and Spatial domain Representation
Can be obtained by a discrete Fourier transform
Frequency Domain Information
The cross-correlation of the signal with itself over varying intervals of time or space
Autocorrelation
The most common enhancement processing approach of the input signal in the time or space domain
Filtering
It is a linear transformation of input samples
Linear Filter
A filter that uses only previous samples of the input or output signals
Causal Filters
A filter that uses only future input samples
Non-causal
Means to change a non-causal filter to a causal filter
Adding a delay
A filter that has constant properties over time
Time-invariant Filter
A filter that changes in time
Adaptive Filter
A filter that produces an output that converges to a constant value with time or remains bounded within a finite interval
Stable Filter
A filter that produces output that diverges
Unstable
A filter that is always stable and only uses the input signals
Finite Impulse Response (FIR) Filter
A filter that uses both the input signal and previous samples of the output signal and can be unstable
Infinite Impulse Response (IIR) Filter
Converts the signal information to a magnitude and phase component of each frequency
Fourier Transform
A function that satisfies the condition:
x(t)=x(t+To)
where,
t = time
To = a constant
Periodic Signal
Any signal for which there is no value of To to satisfy the equation
x(t)=x(t+To)
Nonperiodic or Aperiodic Signal
A signal where there is no uncertainty with respect to ts value at any time
Deterministic Signal
A signal about which there is uncertainty before its actual occurrence
Random Signal
The condition of an energy signal
It is an energy signal IF AND ONLY IF the total energy of the signal satisfies the condition
0
The condition of a power signal
It is a power signal IF AND ONLY IF the average power of the signal satisfies the condition
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It is function where there is a single 1 output and 0 elsewhere
δ(n) = 1
Impulse Function or δ Function
It is a function where it is equal to 1 for n≥0 and 0 elsewhere
u(n) = 1
Unit Step Function
It is a function where it is equal to 1 for n≤ -1 and 0 elsewhere
u(-n-1) = 1
Reversed Step Function
Formula for a sinusoidal signal
x(t) = Acos(ωt + ϕ) x(t) = Acos(2πft + ϕ)
ω is in rad
f is in Hertz
ϕ is the phase angle
A is the dc level
General expression for complex signal
x(t) = Ae^±(ωt + θ) x(t) = cos(ωt + θ) ± jsin(ωt + θ)
The Fourier Series Expansion formula
x(t) = a₀ + 2∑[aᵢcos((2πit)/T₀) + bᵢsin((2πit)/T₀)]
i is from 1 to ∞
where,
aᵢ and bᵢ = coefficients
i/T₀ = ith harmonic of the fundamental frequency f₀
a₀ = mean value of the periodic signal v(t)
Fourier series, Average (a₀) or Mean value of a periodic signal
a₀ = 1/T₀ ∫ x(t) dt
from -T₀/2 to T₀/2
Fourier series, cosine coefficient aᵢ
aᵢ = 1/T₀ ∫x(t) cos((2πit)/T₀) dt
from -T₀/2 to T₀/2
Fourier series, sine coefficient bᵢ
bᵢ= 1/T₀ ∫x(t) sin((2πit)/T₀) dt
from -T₀/2 to T₀/2
A certain linear operator that maps functions to other functions to other function
Fourier Transform
Decomposes a function into a continuous spectrum of its frequency components
Fourier Transform
Synthesizes a function from its spectrum of frequency components.
Inverse Fourier Transform
Fourier Transform Pair condition from time domain to frequency domain and vv
x(t) = ∫X(f) e^(j2πft) df vv X(f) = ∫ x(t) e^(-j2πft) dt
from -∞ to ∞
Fourier Transform Property of Linearity or Superposition from time domain to frequency domain and vv
ax₁(t) + bx₂(t) vv aX₁(f) + bX₂(f)
Fourier Transform Property for Duality from time domain to frequency domain and vv
X(t) vv x(-f)
Fourier Transform Property for Time Scaling from time domain to frequency domain and vv
x(at) vv 1/a x(f/a)
Fourier Transform Property for Time Shifting from time domain to frequency domain and vv
x(t-t₀) vv X(f)e^(-2jπft₀)
Fourier Transform Property for Frequency Shifting from time domain to frequency domain and vv
x(t)e^(2jπfct) vv X(f-fc)
Fourier Transform Property for Differentiation in the Time Domain from time domain to frequency domain and vv
dⁿ/dtⁿ x(t) vv (2πft)ⁿX(f)
Fourier Transform Property for Integration in the Time Domain from time domain to frequency domain and vv
∫x(t) dt vv 1/(2πf) X(f)
from -∞ to t
Fourier Transform Property for Multiplication in the Time Domain from time domain to frequency domain and vv
x₁(t)x₂(t) vv ∫X₁(λ)X₂(f-λ) dλ
from -∞ to t
The multiplication of two signals in the same time domain is transformed into the convolution of their individual Fourier transforms in the frequency domain
Multiplication Theorem
Fourier Transform Property for Convolution in the Time Domain from time domain to frequency domain and vv
∫x₁(τ)x₂(t-τ) dτ vv X₁(f)X₂(f)
from -∞ to t
The convolution of two signals in the time domain is transformed into the multiplication of their individual Fourier Transforms in the Frequency domain
Convolution Theorem
The operation used to measure the similarity between two signals or functions and the time relation of the similarity
Correlation
Steps to correlate two signals
(1) First shift the second signal
(2) Then multiply both signals
(3) Integrate under the curve
When the signals x₁(t) and x₂(t) are the same during the correlation process
Autocorrelation
Provides information about the time-domain structure of a noisy signal
Autocorrelation
Often used to discover periodic components in noisy signals
Autocorrelation
When the signals x₁(t) and x₂(t) are different during the correlation process
Cross correlation
Used to identify signal by comparison with a library of known reference signals
Cross Correlation
It is similar to correlation except that the second signal x(t-τ) is flipped back to front
Convolution
Steps for the convolution operation
(1) Flip the second signal
(2) Shift the second signal
(3) Multiply both signals
(4) Integrate under the curve
A filter whose response to an impulse ultimately settles to zero.
Finite Impulse Response (FIR) Filter
A filter which have internal feedback and may continue to respond indefinitely
Infinite Impulse Response (IIR) Filter
Properties of FIR filters
Inherently Stable
Requires no Feedback
Can have linear phase
Can have minimum phase
Generally linear design methods
Efficient realizations with Fast Fourier Transforms
Startup Transients are finite in duration
Less sensitive to coefficient word lengths
A FIR filter design method where ideal impulse response is truncated with a window. The simplest method and suitable for short filters.
Window Design Method
A FIR filter design method uses the transition region to control sidelobes which specifies response at evenly-spaced frequency samples.
Frequency Sampling Method
A FIR filter design method minimizes the maximum ripple using the Remez exchange algorithm.
Equiripple Design Method or Minimax Method
An impulse response function that is non-zero over an infinite length of time.
Infinite Impulse Response (IIR) Filter
Examples of IIR filters
Chebyshev
Butterworth Filter
Bessel Filter
Elliptic Filter
This IIR filter method transforms filters into discrete-time domain using bilinear transform
Transforms of Continuous Filter Designs
Uses Yule-Walker Method for this IIR Filter modeling technique
Modeling of Desired Frequency Response
Uses Prony’s Method for this IIR modeling technique
Modeling of Desired Impulse Response
The bandwidth, sampling rate, no.of bits, and data rate of high fidelity CD music
5 Hz to 20 kHz
41. kHz
16 bits
706 kbps
The bandwidth, sampling rate, no.of bits, and data rate of tele-phone quality speech
200 Hz to 3.2 kHz
8 kHz
12 bits
96 kbps
The bandwidth, sampling rate, no.of bits, and data rate for telephone with companding
200 Hz to 3.2 kHz
8 kHz
8 bits
64 bits
The bandwidth, , sampling rate, no.of bits, and data rate of speech encoded by linear predictive coding
200 Hz to 3.2 kHz
8 kHz
12 bits
4 kbps
The data rate that represents the straightforward application of sampling and quantization theory to audio signals
64 kbps
It has a shiny main surface and information is burned on the surface with a laser
Compact Disc (CD) or High Fidelity Audio
An audio processing application based on the model of human speech production
Speech Synthesis and Recognition
A data compression application where each time a zero is encountered in the input data, two values are written to the output file.
Run-Length Encoding
A data compression method which assign frequently used character fewer bits, and seldom used characters more bits.
Huffman Encoding
A data compression method that refers to several techniques that store data as the difference between successive samples, rather than directly storing the samples themselves
Delta Encoding
A data compression method which is the foremost technique for general purpose data compression due to its simplicity and versatility. This is always used in GIF image files.
LZW Compression
JPEG
Joint Photographers Experts Group
Another name for JPEG
Transform Compression
Data compression where the low frequency components of a signal are more important than the high frequency components
JPEG
MPEG
Moving Picture Experts Group
It is a compression standard for digital video sequence and also provides for the compression of the sound track associated with the video
MPEG
Two MPEG types
Within-the-frames
Between-frame
A MPEG type where individual frames making up the video sequence are encoded as they were originally still images. This is performed using JPEG standard.
Within-the-Frame
