Fundamental Concepts of Digital Signal Processing

To perform signal processing operations, digital signal processing is used by computers or advanced digital signal processor. Digital signal processing is actually sub-category of Signal processing in addition to the other sub-category analog signal processing.

Signals and systems:
  • Signal: Signal is interpretation how one parameter varies with another variable.
  • System: System is a process in which output signal is gained in response to an input signal.
System in mathematical form:

Moving System= Sum of samples/ No. Of samples

Operations:

Here are 3 operations:

  1. Flipping
  2. Shifting
  3. Scaling

Here we are taking an example to apply these operations having n on x-axis.

1- Flipping:

2- Shifting:
Backward Shifting
Backward Shifting
Forward Shifting
Forward Shifting
3- Scaling:
Shrinking
Shrinking
Expanding
Expanding
Methods to recover original signal from expanded signals:
  • Zero Order hold: Means that in the missing or unknown positions we interpolate the same value until the next sample.
  • 1st Order Hold: In this, linear function is used to interpolate (line with a slope).
  • 2nd Order Hold: In this, three samples are used to recover original signal.
Representation of Infinite signals via delta:

Plots are given for the representation of infinite signals via delta having n on x-axis.

Even and Odd Decomposition:

Energy signal x(t) is expressed as the sum of odd and even parts that can be decomposed through even and odd decomposition.

Example:

Put values of n from -8 to +8 in the even decomposition formula, we get plot having n on x-axis:

Put values of n from -8 to +8 in the odd decomposition formula, we get plot having n on x-axis:

Interconnection of Systems:
1- Series/Cascaded Interconnection:

Mathematical Form:

2- Parallel System:

Mathematical Form:

3- Feedback System:

Mathematical Form:

Properties of Systems:

  • Causal/Non-Causal Systems
  • Linearity
  • Time invariance

1- Causal/Non-Causal Systems:

  • Causal Systems: If output is dependent on present/past values of input then system is causal system.

Example:  y(n)=x(n)-2x(n-1)

  • Non-Causal Systems:

If output is dependent on future values of input then system is non-causal system.

Example:  y(n)=x(n+3)

  • Special Case:

In this you can see this system has future, present and past values. So this system is neither causal nor non-causal system.

2- Linearity:

If the output of scaled sum of two input signals is equal to scaled sum of two output signals then system is said to be linear system.

To show linearity of a system, we have to prove superposition property. Additive and homogeneous property are collectively called superposition property. Lets see an example:

Example:

y(n)=x(n)-2x(n-1)

Now, we will check it is showing superposition property or not.

  • Additive:

So, System is showing additive property.

  • Homogeneous Property:

System is showing homogeneous property. So, system is linear system.

3- Time Invariance:

Time delay or advance in input causes same time shift in output.

Example:

System Representation using its impulse response:
Convolution using tabular method:

Convolution is the important technique of digital signal processing because it creates relation between input, output signals and impulse response.

Here are some fundamental concepts of Digital Signal Processing and you can see more here too.

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