Convolution of discrete and continuous time signals

Therefore, it is possible to convolve a discrete-time signal with a continuous-time one by treating each sample as a value multiplied by the delta function and then summing them together. However, this approach may not always be mathematically rigorous and could lead to errors. In summary, it is possible to convolve a discrete-time signal with a continuous-time one, but it must be done carefully to avoid potential errors.
  • #1
swuster
42
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Not a specific question per se but...

Is it possible to convolve a discrete-time signal with a continuous-time one?

if you have x(n) and y(t) can you calculate the convolution of x and y (say, by taking y(t) for t in the set of integers or by treating each x(n) as its value multiplied by the delta function) or is there something inherently wrong about this?
 
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  • #2
A discrete-time signal x(nT), where T is the sampling period, is a special case of a continuous-time signal, as you can write

[tex]x(t)=\sum_{n}x(nT)\delta(t-nT)[/tex]

so if y(t) is a continuous-time signal, the convolution is

[tex]x(t)\otimes y(t)=\sum_{n}x(nT)\delta(t-nT)\otimes y(t)=\sum_{n}x(nT)y(t-nT)[/tex]

(you have to assume this converges).
 
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1. What is the difference between discrete and continuous time signals?

Discrete time signals are sampled at specific time intervals, while continuous time signals are measured at every point in time. In other words, discrete time signals have a finite number of values, while continuous time signals have an infinite number of values.

2. How is convolution performed on discrete time signals?

Convolution of discrete time signals involves multiplying each value of one signal by all the values of the other signal, and then summing up the products. This process is repeated for every sample in the signals.

3. Can convolution be performed on continuous time signals?

Yes, convolution can be performed on continuous time signals by converting them into discrete time signals through sampling. The resulting discrete time signals can then be convolved using the same process as discrete time signals.

4. What is the significance of convolution in signal processing?

Convolution allows us to combine two signals to obtain a new signal that contains information from both signals. This is useful in filtering, noise reduction, and other signal processing applications.

5. Are there any limitations to convolution of discrete and continuous time signals?

One limitation is that convolution assumes linearity and time-invariance in the signals, which may not always hold true in real-world scenarios. Additionally, the size of the resulting convolved signal may be larger than the original signals, which can be a computational challenge for large signals.

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