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Page 1 ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete Objectives: Representation of DT Signals Response of DT LTI Systems Convolution Examples Properties CONVOLUTION OF DISCRETE-TIME SIGNALS URL: Page 2 ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete Objectives: Representation of DT Signals Response of DT LTI Systems Convolution Examples Properties CONVOLUTION OF DISCRETE-TIME SIGNALS URL: EE 3512: Lecture 14, Slide 1 Are there sets of “basic” signals, x k [n], such that: § We can represent any signal as a linear combination (e.g, weighted sum) of these building blocks? (Hint: Recall Fourier Series.) § The response of an LTI system to these basic signals is easy to compute and provides significant insight. For LTI Systems (CT or DT) there are two natural choices for these building blocks: § Later we will learn that there are many families of such functions: sinusoids, exponentials, and even data-dependent functions. The latter are extremely useful in compression and pattern recognition applications. Exploiting Superposition and Time-Invariance DT LTI System å = k k k n x a n x ] [ ] [ å = k k k n y b n y ] [ ] [ § DT Systems: (unit pulse) § CT Systems: (impulse) ( ) 0 t t - d [ ] 0 n n - d Page 3 ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete Objectives: Representation of DT Signals Response of DT LTI Systems Convolution Examples Properties CONVOLUTION OF DISCRETE-TIME SIGNALS URL: EE 3512: Lecture 14, Slide 1 Are there sets of “basic” signals, x k [n], such that: § We can represent any signal as a linear combination (e.g, weighted sum) of these building blocks? (Hint: Recall Fourier Series.) § The response of an LTI system to these basic signals is easy to compute and provides significant insight. For LTI Systems (CT or DT) there are two natural choices for these building blocks: § Later we will learn that there are many families of such functions: sinusoids, exponentials, and even data-dependent functions. The latter are extremely useful in compression and pattern recognition applications. Exploiting Superposition and Time-Invariance DT LTI System å = k k k n x a n x ] [ ] [ å = k k k n y b n y ] [ ] [ § DT Systems: (unit pulse) § CT Systems: (impulse) ( ) 0 t t - d [ ] 0 n n - d EE 3512: Lecture 14, Slide 2 Representation of DT Signals Using Unit Pulses Page 4 ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete Objectives: Representation of DT Signals Response of DT LTI Systems Convolution Examples Properties CONVOLUTION OF DISCRETE-TIME SIGNALS URL: EE 3512: Lecture 14, Slide 1 Are there sets of “basic” signals, x k [n], such that: § We can represent any signal as a linear combination (e.g, weighted sum) of these building blocks? (Hint: Recall Fourier Series.) § The response of an LTI system to these basic signals is easy to compute and provides significant insight. For LTI Systems (CT or DT) there are two natural choices for these building blocks: § Later we will learn that there are many families of such functions: sinusoids, exponentials, and even data-dependent functions. The latter are extremely useful in compression and pattern recognition applications. Exploiting Superposition and Time-Invariance DT LTI System å = k k k n x a n x ] [ ] [ å = k k k n y b n y ] [ ] [ § DT Systems: (unit pulse) § CT Systems: (impulse) ( ) 0 t t - d [ ] 0 n n - d EE 3512: Lecture 14, Slide 2 Representation of DT Signals Using Unit Pulses EE 3512: Lecture 14, Slide 3 Response of a DT LTI Systems – Convolution Define the unit pulse response, h[n], as the response of a DT LTI system to a unit pulse function, d[n]. Using the principle of time-invariance: Using the principle of linearity: Comments: § Recall that linearity implies the weighted sum of input signals will produce a similar weighted sum of output signals. § Each unit pulse function, d[n-k], produces a corresponding time-delayed version of the system impulse response function (h[n-k]). § The summation is referred to as the convolution sum. § The symbol “*” is used to denote the convolution operation. DT LTI å = k k k n x a n x ] [ ] [ å = k k k n y b n y ] [ ] [ [ ] n h ] [ ] [ ] [ ] [ k n h k n n h n - ® - Þ ® d d ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ n h n x k n h k x n y k n k x n x k k * = - = ® - = å å ¥ -¥ = ¥ -¥ = d convolution sum convolution operator Page 5 ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete Objectives: Representation of DT Signals Response of DT LTI Systems Convolution Examples Properties CONVOLUTION OF DISCRETE-TIME SIGNALS URL: EE 3512: Lecture 14, Slide 1 Are there sets of “basic” signals, x k [n], such that: § We can represent any signal as a linear combination (e.g, weighted sum) of these building blocks? (Hint: Recall Fourier Series.) § The response of an LTI system to these basic signals is easy to compute and provides significant insight. For LTI Systems (CT or DT) there are two natural choices for these building blocks: § Later we will learn that there are many families of such functions: sinusoids, exponentials, and even data-dependent functions. The latter are extremely useful in compression and pattern recognition applications. Exploiting Superposition and Time-Invariance DT LTI System å = k k k n x a n x ] [ ] [ å = k k k n y b n y ] [ ] [ § DT Systems: (unit pulse) § CT Systems: (impulse) ( ) 0 t t - d [ ] 0 n n - d EE 3512: Lecture 14, Slide 2 Representation of DT Signals Using Unit Pulses EE 3512: Lecture 14, Slide 3 Response of a DT LTI Systems – Convolution Define the unit pulse response, h[n], as the response of a DT LTI system to a unit pulse function, d[n]. Using the principle of time-invariance: Using the principle of linearity: Comments: § Recall that linearity implies the weighted sum of input signals will produce a similar weighted sum of output signals. § Each unit pulse function, d[n-k], produces a corresponding time-delayed version of the system impulse response function (h[n-k]). § The summation is referred to as the convolution sum. § The symbol “*” is used to denote the convolution operation. DT LTI å = k k k n x a n x ] [ ] [ å = k k k n y b n y ] [ ] [ [ ] n h ] [ ] [ ] [ ] [ k n h k n n h n - ® - Þ ® d d ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ n h n x k n h k x n y k n k x n x k k * = - = ® - = å å ¥ -¥ = ¥ -¥ = d convolution sum convolution operator EE 3512: Lecture 14, Slide 4 LTI Systems and Impulse Response The output of any DT LTI is a convolution of the input signal with the unit pulse response: Any DT LTI system is completely characterized by its unit pulse response. Convolution has a simple graphical interpretation: DT LTI ] [n x ] [ * ] [ ] [ n h n x n y = [ ] n h ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ n h n x k n h k x n y k n k x n x k k * = - = ® - = å å ¥ -¥ = ¥ -¥ = dRead More
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1. What is discrete time convolution? |
2. How is discrete time convolution different from continuous time convolution? |
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5. Are there any properties associated with discrete time convolution? |
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