Advanced Digital Signal Processing Introduction LECTUREI 1 DiscreteTime

Advanced Digital Signal Processing -Introduction LECTURE-I 1

Discrete-Time Signals The entire subject is focused on discrete time signals, which usually are A/D converted signals 2

Discrete-Time Signals • An important signal is the unit sample • It can be used to represent other discrete time signals 3

Discrete-Time Signals • The unit step • Is related to the unit sample by • A complex exponential is important for Fourier decomposition of signals 4

Discrete-Time Systems Any system that maps an input sequence to an output sequence 5

Discrete-Time Systems • For a linear system, this must hold, where a and b are arbitrary constants • Note that this implies, that if the input is written as sum over unit impulses • the output is • where 6

Discrete-Time Systems • For a shift-invariant system, if the input is shifted by n 0 then the output must also shift by n 0. I. e. , the system does not change over time • For a system that is both linear and shift invariant (LSI) the following holds • This is called the convolution sum, which is also written as • Hence an LSI system is completely specified by its unit sample response h(n) 7

Discrete-Time Systems • A system is called causal if at any time n 0 the output of the system only depends on input values for n<=n 0 • An LSI system can only be causal if h(n)=0 for n<0 • A LSI system is called stable in the Bounded-Input Bounded-Output sense if 8

Time-Domain Description of LSI Filters • An important class of linear shift-invariant systems are those whose input x(n) and output y(n) are related by a linear constant coefficient difference equation • p and q determine the order of the system and a(k) and b(k) are the filter coefficients. This is often written in this form 9

Time-Domain Description of LSI Filters • If p=0, then the following form result and hence the unit sample response h(n) is finite in length, hence these filters are called finite impulse response filter (FIR) • If p is not zero then the impulse response is not finite hence the filters are called IIR filters 10

Discrete-Time Fourier Transform • The discrete-time Fourier transform (DTFT) is defined as • A sufficient condition that this sum converges is 11

Discrete-Time Fourier Transform • The frequency response of a LTI filter is the DTFT of the unit sample response • The DTFT is invertible 12

Discrete-Time Fourier Transform • The convolution theorem is quite helpful • Another useful theorem is Parseval’s theorem 13

Discrete-Time Fourier Transform 14

The z-Transform • A generalization to the DTFT, allowing many signals without DTFT to be transformed • Note that for z=ejw holds • Note that the sum might not converge for all possible z, hence there is always a region of convergence associated with a z transform 15

The z-Transform • When the z-transform is used for power spectral densities, then the following property comes in handy 16

The z-Transform 17

The z-Transform 18

The z-Transform • The z-transform of the unit sample response of a LTI system will become important later on • For FIR filters this becomes and the roots of this polynomial are called the zeros of the filter 19

The z-Transform • For IIR filters this becomes and the roots of the denominator polynomial are called the poles of the filter • If the coefficients are real-valued (equivalently h(n) is real valued) then and all poles and zeros will occur in conjugate pairs 20

The z-Transform 21

Special Classes of Filters • Generalized linear phase filters have the following frequency response, where A(ejw) is a real valued function and a and b are constants • This can be achieved with FIR filters having either conjugate symmetric (Hermitian) or conjugate anti symmetric (anti. Hermitian) unit sample responses • This though implies in other words, if H(z) has a zero at z 0 then H(z) must also have a zero at 1/z 0* 22

Special Classes of Filters • All pass filters are another special class of filters, which have a frequency response with constant magnitude • An all pass filter having a rational system function H(z) must be of the following form • Hence if H(z) has a pole at ak, then H(z) also has a pole at 1/ak* 23