340 likes | 612 Views
Review of DSP. Signal and Systems:. Signal are represented mathematically as functions of one or more independent variables. Digital signal processing deals with the transformation of signal that are discrete in both amplitude and time.
E N D
Signal and Systems: • Signal are represented mathematically as functions of one or more independent variables. • Digital signal processing deals with the transformation of signal that are discrete in both amplitude and time. • Discrete time signal are represented mathematically as sequence of numbers.
Signals and Systems: • A discrete time system is defined mathematically as a transformation or operator. • y[n] = T{ x[n] } T{.} x [n] y [n]
Linear Systems: • The class of linear systems is defined by the principle of superposition. • And • Where a is the arbitrary constant. • The first property is called the additivity property and the second is called the homogeneity or scaling property.
Linear Systems: • These two property can be combined into the principle of superposition, H Linear System H H
Time-Invariant Systems: • A Time-Invariant system is a system for witch a time shift or delay of the input sequence cause a corresponding shift in the output sequence. H H
LTI Systems: • A particular important class of systems consists of those that are linear and time invariant. • LTI systems can be completely characterized by their impulse response. • From principle of superposition: • Property of TI:
LTI Systems (Convolution): • Above equation commonly called convolution sum and represented by the notation
Convolution properties: • Commutativity: • Associativity: • Distributivity: • Time reversal:
H1 H2 H2 H1 …Convolution properties: • If two systems are cascaded, • The overall impulse response of the combined system is the convolution of the individual IR: • The overall IR is independent of the order:
Duration of IR: • Infinite-duration impulse-response (IIR). • Finite-duration impulse-response (FIR) • In this case the IR can be read from the right-hand side of:
Transforms: • Transforms are a powerful tool for simplifying the analysis of signals and of linear systems. • Interesting transforms for us: • Linearity applies: • Convolution is replaced by simpler operation:
…Transforms: • Most commonly transforms that used in communications engineering are: • Laplace transforms (Continuous in Time & Frequency) • Continuous Fourier transforms (Continuous in Time) • Discrete Fourier transforms (Discrete in Time) • Z transforms (Discrete in Time & Frequency)
The Z Transform: • Definition Equations: • Direct Z transform • The Region Of Convergence (ROC) plays an essential role.
The Z Transform (Elementary functions): • Elementary functions and their Z-transforms: • Unit impulse: • Delayed unit impulse:
The Z Transform (…Elementary functions): • Unit Step: • Exponential:
Z Transform (Cont’d) • Important Z Transforms Region Of Convergence (ROC) Whole Page Whole Page Unit Circle |z| > |a|
The Z Transform (Elementary properties): • Elementary properties of the Z transforms: • Linearity: • Convolution: if ,Then
The Z Transform (…Elementary properties): • Shifting: • Differences: • Forward differences of a function, • Backward differences of a function,
The Z Transform (…Region Of Convergence for Z transform): • Since the shifting theorem
The Z Transform (Region Of Convergence for Z transform): • The ROC is a ring or disk in the z-plane centered at the origin :i.e., • The Fourier transform of x[n] converges at absolutely if and only if the ROC of the z-transform of x[n] includes the unit circle. • The ROC can not contain any poles.
The Z Transform (…Region Of Convergence for Z transform): • If x[n] is a finite-duration sequence, then the ROC is the entire z-plane, except possibly or . • If x[n] is a right-sided sequence, the ROC extends outward from the outermost finite pole in to . • The ROC must be a connected region.
The Z Transform (…Region Of Convergence for Z transform): • A two-sidedsequence is an infinite-duration sequence that is neither right sided nor left sided. • If x[n] is a two-sided sequence, the ROC will consist of a ring in the z-plane, bounded on the interior and exterior by a pole and not containing any poles. • If x[n] is a left-sided sequence, the ROC extends in ward from the innermost nonzero pole in to .
The Z Transform (Application to LTI systems): • We have seen that • By the convolution property of the Z transform • Where H(z) is the transfer function of system. • Stability • A system is stable if a bounded input produced a bounded output, and a LTI system is stable if:
Fourier Transform Time Transform Type Frequency Continuous Continuous Fourier Transform Continuous Discrete DiscreteTime Continuous FFT Fourier Series Discrete Continuous Discrete Discrete Discrete Time Discrete FFT
The Discrete Fourier Transform (DFT) • Definition Equations: • Direct Z transform • It is customary to use the • Then the direct form is:
The Discrete Fourier Transform (DFT) • With the same notation the inverse DFT is
The DFT (Elementary functions): • Elementary functions and their DFT: • Unit impulse: • Shifted unit impulse:
The DFT (…Elementary functions): • Constant: • Complex exponential:
The DFT (…Elementary functions): • Cosine function:
The DFT (Elementary properties): • Elementary properties of the DFT: • Symmetry: If ,Then • Linearity: if and ,Then
The DFT (…Elementary properties): • Shifting: because of the cyclic nature of DFT domains, shifting becomes a rotation. if ,Then • Time reversal: if ,Then
The DFT (…Elementary properties): • Cyclic convolution: convolution is a shift, multiply and add operation. Since all shifts in the DFT are circular, convolution is defined with this circularity included.