5th Lecture SP.pdf

Signal Processing FIFth Lecture Md. Toukir Ahmed Lecturer, Dept. Of IRE, BDU. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 1 Digital Signal Processing - Static Systems: Some systems have feedback and some do not. Those, wh...

Signal Processing FIFth Lecture Md. Toukir Ahmed Lecturer, Dept. Of IRE, BDU. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 1 Digital Signal Processing - Static Systems: Some systems have feedback and some do not. Those, which do not have feedback systems, their output depends only upon the present values of the input. Past value of the data is not present at that time. These types of systems are known as static systems. It does not depend upon future values too. Since these systems do not have any past record, so they do not have any memory also. Therefore, we say all static systems are memory-less systems. Let us take an example to understand this concept much better. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 2 Example: Let us verify whether the following systems are static systems or not. 1. y(t)=x(t)+x(t−1) 2. y(t)=x(2t) 3. y(t)=x=sin[x(t)] 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 3 a) y(t)=x(t)+x(t−1) Here, xt is the present value. It has no relation with the past values of the time. So, it is a static system. However, in case of xt−1, if we put t = 0, it will reduce to x−1 which is a past value dependent. So, it is not static. Therefore here yt is not a static system. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 4 b) y(t)=x(2t) If we substitute t = 2, the result will be yt= x4. Again, it is future value dependent. So, it is also not a static system. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 5 c) y(t)=x=sin[x(t)] In this expression, we are dealing with sine function. The range of sine function lies within -1 to +1. we can say it is not dependent upon any past or future values. Hence, it is a static system. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 6  From the examples, we can draw the following conclusions − 1. Any system having time shifting is not static. 2. Any system having amplitude shifting is also not static. 3. Integration and differentiation cases are also not static. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 7 Digital Signal Processing - Dynamic Systems: If a system depends upon the past and future value of the signal at any instant of the time then it is known as dynamic system. Dynamic Systems, these are not memory less systems. They store past and future values. Therefore, they require some memory. Let us understand this theory better through some examples. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 8 Examples: Find out whether the following systems are dynamic. a) y(t)=x(t+1) In this case if we put t = 1 in the equation, it will be converted to x2, which is a future dependent value. Because here we are giving input as 1 but it is showing value for x2. As it is a future dependent signal, so clearly it is a dynamic system. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 9 b) y(t)=cos[x(t)] In this case, as the system is cosine function it has a certain domain of values which lies between -1 to +1. Therefore, whatever values we will put we will get the result within specified limit. Therefore, it is a static system. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 10 From the examples, we can draw the following conclusions − 1) All time shifting cases signals are dynamic signals. 2) In case of time scaling too, all signals are dynamic signals. 3) Integration cases signals are dynamic signals. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 11 Digital Signal Processing - Causal Systems: Previously, we saw that the system needs to be independent from the future and past values to become static. In this case, the condition is almost same with little modification. Here, for the system to be causal, it should be independent from the future values only. That means past dependency will cause no problem for the system from becoming causal. Causal systems are practically or physically realizable system. Let us consider some examples to understand this much better. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 12 Examples: Let us consider the following signals. a) y(t)=x(t) Here, the signal is only dependent on the present values of x. For example if we substitute t = 3, the result will show for that instant of time only. Therefore, as it has no dependence on future value, we can call it a Causal system. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 13 b) y(t)=x(t−1) Here, the system depends on past values. For instance if we substitute t = 3, the expression will reduce to x2, which is a past value against our input. At no instance, it depends upon future values. Therefore, this system is also a causal system. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 14 c) y(t)=x(t)+x(t+1) In this case, the system has two parts. The part xt, as we have discussed earlier, depends only upon the present values. So, there is no issue with it. However, if we take the case of xt+1, it clearly depends on the future values because if we put t = 1, the expression will reduce to x2 which is future value. Therefore, it is not causal. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 15 For next tutorial class, we will discuss the below topics: i) DSP - Non-Causal Systems (https://www.tutorialspoint.com/digital_signal_processing/dsp_non_causal_systems.htm) ii) DSP - Anti-Causal Systems (https://www.tutorialspoint.com/digital_signal_processing/dsp_anti_causal_systems.htm) 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 16 Digital Signal Processing - Linear Systems: The system is a combination of two types of laws − 1. Law of additivity 2. Law of homogeneity 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 17 However, there are some other conditions to check whether the system is linear or not. The conditions are −  The output should be zero for zero input.  There should not be any non-linear operator present in the system. Examples of non-linear operators − a. Trigonometric operators- Sin, Cos, Tan, Cot, Sec, Cosec etc. b. Exponential, logarithmic, modulus, square, Cube etc. c. Either input x or output y should not have these non-linear operators. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 18 Examples: Let us find out whether the following systems are linear. a) y(t)=x(t)+3 This system is not a linear system because it violates the first condition 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 19 Examples: b) y(t)=sin(x(t)) In the above system, first condition is satisfied because if we put xt = 0, the output will also be sin0 = 0. However, the second condition is not satisfied, as there is a non-linear operator which operates xt. Hence, the system is not linear. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 20 For next tutorial class, we will discuss the below topic: i) DSP - Non-Linear Systems (https://www.tutorialspoint.com/digital_signal_processing/dsp_non_linear_systems.htm) 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 21 Digital Signal Processing - Stable Systems: A stable system satisfies the BIBO (Bounded Input for Bounded Output) condition. Here, bounded means finite in amplitude. For a stable system, output should be bounded or finite, for finite or bounded input, at every instant of time. Some examples of bounded inputs are functions of sine, cosine, unit step. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 22 Examples: a) y(t)=x(t)+10 Here, for a definite bounded input, we can get definite bounded output i.e. if we put x(t)=2,y(t)=12, which is bounded in nature. Therefore, the system is stable. b) y(t)=sin[x(t)] In the given expression, we know that sine functions have a definite boundary of values, which lies between -1 to +1. So, whatever values we will substitute at xt, we will get the values within our boundary. Therefore, the system is stable. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 23 For next tutorial class, we will discuss the below topic: i) Digital Signal Processing - Unstable Systems (https://www.tutorialspoint.com/digital_signal_processing/dsp_unstable_systems.htm) ii) DSP - Time-Invariant Systems (https://www.tutorialspoint.com/digital_signal_processing/dsp_time_invariant_systems.htm) iii) DSP - Time-Variant Systems (https://www.tutorialspoint.com/digital_signal_processing/dsp_time_variant_systems.htm) 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 24  Source: https://www.tutorialspoint.com/digital_signal_processing/dsp_static_syste ms.htm https://www.tutorialspoint.com/digital_signal_processing/dsp_dynamic_s ystems.htm https://www.tutorialspoint.com/digital_signal_processing/dsp_causal_syst ems.htm https://www.tutorialspoint.com/digital_signal_processing/dsp_linear_syst ems.htm https://www.tutorialspoint.com/digital_signal_processing/dsp_stable_syst ems.htm https://www.tutorialspoint.com/digital_signal_processing/dsp_system_pro perties_solved_examples.htm 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 25  Thank You. 9/13/2024 ICT 4355: Signal Processing (5th Lecture) 26