Convolution in Discrete time of a function with Impulse with delay?

In summary, convolution in discrete time is a mathematical operation that combines two discrete functions to produce a third function. It is different from continuous time convolution in that it deals with discrete functions and involves simpler computation. An impulse function is a mathematical function with an infinite value at one point, often used to represent an instantaneous event. Delay in convolution refers to shifting the impulse function in time, resulting in a shift in the output function. Convolution in discrete time has various real-world applications, including audio and image processing, systems analysis and control, and scientific and engineering fields.
  • #1
Kdar
3
0
Here is convolution:

c[k]= (0.5)^k * delta(k-1)

What do I do about delta(k-1)?
I know if it is c[k]= (0.5)^k * delta(k), then it just equal (0.5)^k

But what do I do with delta(k-1)?
 
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  • #2
Welcome to PF!

Hi Kdar! Welcome to PF! :smile:

(have a delta: δ and try using the X2 tag just above the Reply box :wink:)

One method is to substitute u = k - 1, solve it, then substitute back again. :smile:
 
  • #3
Why not start from the definition then go with it?
 

1. What is convolution in discrete time?

Convolution in discrete time is a mathematical operation that combines two discrete functions to produce a third function. It is commonly used in signal processing and is used to model the output of a linear time-invariant system when the input is a sequence of impulses.

2. How is convolution in discrete time different from continuous time convolution?

The main difference between convolution in discrete time and continuous time is that discrete time convolution deals with discrete functions and sums them, while continuous time convolution deals with continuous functions and integrates them. Discrete time convolution is also easier to compute as it involves simple multiplication and summation.

3. What is an impulse function?

An impulse function, also known as the Dirac delta function, is a mathematical function that has a value of zero everywhere except at one point, where it has an infinite value. It is often used to represent a single, instantaneous event that has no duration, such as a sudden spike in a signal.

4. How does delay affect the convolution of a function with an impulse?

Delay in convolution refers to shifting the impulse function in time before convolving it with the input function. This results in the output function being shifted in time as well. The amount of delay will determine the amount of shift in the output function. In general, a larger delay will result in a larger shift in the output function.

5. What are some real-world applications of convolution in discrete time?

Convolution in discrete time has many practical applications, such as in audio and image processing, where it is used for filtering, noise reduction, and feature detection. It is also used in systems analysis and control, communication systems, and digital signal processing. Additionally, convolution in discrete time is used in scientific and engineering fields to model and analyze various systems and phenomena.

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