Digital Signal Processing Lecture 9 Review of LTI systems بسم الله الرحمن الرحيم University of Khartoum Department of Electrical and Electronic Engineering Diploma/M. Sc. Program in Telecommunication and Information Systems 2013-2014 Digital Signal Processing Lecture 9 Review of LTI systems Dr. Iman AbuelMaaly

System design and implementation In practice, system design and implementation are usually treated jointly rather than separately. Often, the system design is driven by the method of implementations and implementation constraints such as cost, hardware limitations, size limitations and power requirements. 2013 - 2014

Types of Systems Finite Impulse Response (FIR) Infinite Impulse Response (IIR) Recursive System Non-recursive 2013 - 2014

Finite Impulse Response (FIR) Systems Digital Filters Finite Impulse Response (FIR) Systems FIR systems has a finite memory: The convolution formula reduces to The system acts as a window that views only the most recent input signal samples for forming the ouptut.

Infinite Impulse Response (IIR) Systems Digital Filters Infinite Impulse Response (IIR) Systems An IIR Systems has an infinite duration impulse response.

Recursive systems Recursive systems can be expressed in general as: Where F[.] denotes some function of its arrangements. In recursive systems the response y(n) depends on past output and the present and past inputs.

Recursive systems A simple recursive system described by a first order difference equation: Where a is a constant. This system is a LTI system. Its block diagram realization can be shown as below: 2013 - 2014

An example of a Recursive Filter Recursive systems An example of a Recursive Filter 2013 - 2014

Non-recursive Systems In contrast, if y(n) depends only on the present and past inputs, then, Such a system is called non-recursive system. Finite Impulse Response systems (FIR) have the above form.

An example of nonrecursive system described by a difference equation: Non-recursive Systems An example of nonrecursive system described by a difference equation: 2013 - 2014

An example of nonrecursive system described by a difference equation: Non-recursive Systems An example of nonrecursive system described by a difference equation: 2013 - 2014

The basic form of: a) Causal non recursive system: b) Causal recursive system

Note The feed back which contains a delay elements is crucial in recursive systems. For recursive systems: to compute the output which is excited with an input at time n, you need to compute all the previous values (i.e., y(0), y(1), ..y(n-1)) Whereas for non-recursive systems the output can be calculated in any order [i.e., y(20), y(11),..y(3)]

Structure for the Realization of LTI Systems Consider the first-order system Which is realized below 2013 - 2014

Structure for realization of LTI systems This realization uses separate delays for both the input and the output, and is called the direct form I structure. 2013 - 2014

Structure for realization of LTI systems This system can be viewed as two LTI systems, the first is a non-recursive, system described by The second is a recursive system described by If we can interchange the order of the cascaded LTI systems, the response of the system remain the same 16 2013 - 2014 2009

v(n) - 2013 - 2014

The resulting figure will be And we get 2013 - 2014

The two delay elements contain the same input w(n) and the same output w(n-1) and thus can be merged into one delay as shown below The new design requires only one delay and hence is more efficient in terms of memory requirements. 2013 - 2014

It is called direct form II structure and it is used extensively in practical applications. These structures can be generalized for the general LTI recursive system described by the difference equation 2013 - 2014

The Direct Form I Structure v(n) 2013 - 2014

The Direct Form I structure This structure requires M+N delays and N+M+1 multiplications. It is a cascade of non-recursive system And a recursive system 2013 - 2014

The Direct Form II Structure for N>M 2013 - 2014

x(n) y(n) w(n) b0 -a1 b1 w(n-1) -a2 b2 -a3 b3 -aN-2 bM (M=N-2) -aN-1 w(n-N) 2013 - 2014

The Direct Form II structure This structure is the cascade of the recursive system Followed by the non-recursive system 2013 - 2014

The Direct Form II structure If N>M, the number of delays is equal to N If M>N, the number of delays is equal to M (i.e, the number of delays is the max of {M,N}) In both cases the required number of multiplications is N+M+1 Direct form II is called the canonic form. 2013 - 2014

The general LTI recursive system is described by the difference equation Special cases: ak =0, for k=0, 1,2, .. M=0 2013 - 2014

Then the above equation becomes ak =0, for k=0, 1,2, .. Then the above equation becomes This is a non- recursive system. It is an FIR system with an impulse response 2013 - 2014

Which is a “purely recursive” system (b) M=0, then Which is a “purely recursive” system 2013 - 2014

LTI systems described by a second order difference equation The second order systems are usually used as basic building blocks for realizing higher order systems The most general second-order system is described as follows: If N=M=2 2013 - 2014

The direct form II structure is as follows 2013 - 2014

Structure realization of this system is as follows: Special cases (a) If we set a1= a2=0, then Which is the FIR system. Structure realization of this system is as follows: 2013 - 2014

(b) If we set b1= b2=0, we obtain the purely recursive second-order system described by 2013 - 2014

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