1. Digital Signals and Systems

2. Graphical representation of a discrete-time signal 𝒙(𝒏)
Digital Signals
A discrete-time signal 𝑥(𝑛) is a function of an integer variable 𝑛. In the DS processor, the signal is represented by the discrete encoded values with a finite precision.
Tabular representation
Functional representation
Graphical representation of a discrete-time signal 𝒙(𝒏)
𝑥 𝑛 = …,0 ,𝟎 ,1 ,4 ,1, 0, 0,… 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒−𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑠𝑖𝑔𝑛𝑎𝑙
𝑥 𝑛 = 𝟎 ,−2 ,1 ,4 ,−1, 𝑓𝑖𝑛𝑖𝑡𝑒−𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑠𝑖𝑔𝑛𝑎𝑙
Sequence representation (bold or arrow for origin n=0)
Mathematically a discrete-time signal 𝑥 𝑛 can be determined by

3. Common Digital Sequences
Unit-impulse sequence:
Unit-step sequence:
𝑥 𝑛 = 𝑎 𝑛 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛
Exponential sequence:
0 < a < 1

4. Shifted Sequences Shifted unit-impulse Shifted unit-step
Right shift by
two samples
Left shift by
two samples

5. Example 1 Sketch this following digital signal sequence,
Solution:

6. Generation of Digital Signals
In order to generate the digital sequence from its analog signal function, the analog signal 𝑥 𝑡 is uniformly sampled at the time interval of ∆𝑡=𝑇 , T is the sampling period.
x(n): digital signal
x(t): analog signal
sampling interval : ∆𝑡=𝑇
𝑥 𝑛 = 𝑥(𝑡 𝑡=𝑛𝑇 =𝑥(𝑛𝑇
Also

7. Example 2
Convert analog signal x(t) into digital signal x(n), when sampling period is 125 microsecond, also plot sample values.
Solution:
The first five sample values:
Plot of the digital sequence:

8. Digital Systems 𝒚 𝒏 𝑥 𝑛 Digital System
A digital system is a device or an algorithm that performs prescribed operations (or transformation) on a signal 𝑥 𝑛 , called the input or excitation, to produce another signal 𝑦 𝑛 called the output or response of the system.
Digital System
Input signal
(Excitation)
Output signal
(Response)
𝑥 𝑛
𝒚 𝒏
transformation
Example
Determine the response of the following system (mean value of three values) to the input signal
𝑥 𝑛 = …, 0,3, 2, 1, 𝟎, 1, 2, 3, 0, … 𝑤𝑖𝑡ℎ 𝑥 0 =0 𝑏𝑜𝑙𝑑 .
𝑥 𝑛 = 𝑛 , −3≤𝑛≤3 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝑦 0 = 𝑥 −1 +𝑥 0 +𝑥(1 = = 2 3
𝑦 𝑛 = 𝑥 𝑛+1 +𝑥 𝑛 +𝑥(𝑛−1
𝑦 𝑛 = …, 0,1, 5 3 , 2, 1, 𝟐 𝟑 , 1, 2, 5 3 , 1, 0, …

9. Block Diagram Representation of Discrete-Time Systems
adder
signal multiplier
Constant multiplier
unit delay element
unit advance element

10. Example 3
Sketch the block diagram representation of the discrete-time system described by the input-output relation.
𝑦 𝑛 = 1 4 𝑦 𝑛− 𝑥 𝑛 𝑥(𝑛−1)
Solution
The output 𝑦 𝑛 is obtained by rearranging the input-output equation:
𝑦 𝑛 = 1 4 𝑦 𝑛− 𝑥 𝑛 +𝑥(𝑛−1

11. Causal and non-causal systems
A system is causal if its output at any time 𝑛 depends only on present and past inputs [i.e. 𝑥 𝑛 , 𝑥 𝑛−1 , 𝑥 𝑛−2 , …], but does not depend on future inputs [i.e. 𝑥 𝑛+1 , 𝑥 𝑛+2 , …].
Example 4
Determine whether the systems described below are causal
Solution
Since for 𝑛>0 the output 𝑦 𝑛 depends on the current input 𝑥 𝑛 and its past value 𝑥 𝑛−2 the system is causal.
b. Since for 𝑛>0 the output 𝑦 𝑛 depends on the current input 𝑥 𝑛 and its future value 𝑥 𝑛+1 the system is non-causal.

12. Linear System Time, t Continuous system Sample number, n
discrete system

13. Linear Systems: Property 1
A digital system S is linear if and only if it satisfy the superposition principle (Homogeneity and Additivity):
Homogeneity: (Deals with amplitude )
If x[n]  y[n], then kx[n]  ky[n]
K is a constant

14. Linear Systems: Property 2
Additivily
Homogeneity & Additivity
(superposition principle)

15. Example 5 (a) Let a digital amplifier, If the inputs are:
Outputs will be:
If we apply combined input to the system:
The output will be:
Individual outputs:
X 10
2 𝑥 1 𝑛 =2𝑢(𝑛)
2 𝑦 1 𝑛 =20𝑢(𝑛)

16. Example 5 (b) System 𝑥 𝑛 𝑦 𝑛 = 𝑥 2 (𝑛) System 𝑥 1 𝑛 =𝑢(𝑛)
𝑥 1 𝑛 =𝑢(𝑛)
𝑦 1 𝑛 = 𝑥 2 (𝑛)
System
𝑥 2 𝑛 =𝛿(𝑛)
𝑦 2 𝑛 = 𝛿 2 (𝑛)
If the input is:
4 𝑥 1 𝑛 +2 𝑥 2 (𝑛)
𝑦 𝑛 = 𝑥 2 (𝑛) = ( 4 𝑥 1 𝑛 +2 𝑥 2 (𝑛)) 2
= (4𝑢 𝑛 +2𝛿(𝑛)) 2 =16 𝑢 2 𝑛 +16𝑢 𝑛 𝛿 𝑛 +4 𝛿 2 𝑛
=16𝑢 𝑛 +20𝛿 𝑛 .
Then the output is:

17. Linear Systems: Property 3
Shift (time) Invariance
A system is called time-invariant (or shift-invariant) if its input-output characteristics do not change with time (a shifted input signal will produce a shifted output signal, with the same of shifting amount).

18. Example 6 (a)
Given the linear system 𝑦 𝑛 =2𝑥 𝑛−5 , find whether the system is time invariant or not.
Solution:
System
𝑥 1 𝑛
𝑦 1 𝑛 =2 𝑥 1 (𝑛−5)
Let the shifted input be:
𝑥 2 𝑛 = 𝑥 1 𝑛− 𝑛 0
Equal
Therefore system output:
𝑦 2 𝑛 =2 𝑥 2 𝑛−5 =2 𝑥 1 (𝑛− 𝑛 0 −5)
Shifting
𝑦 2 𝑛 =2 𝑥 1 (𝑛−5)
By 𝑛 0 samples leads to 𝑦 1 𝑛− 𝑛 0 =2 𝑥 1 (𝑛−5− 𝑛 0 )

19. Example 6 (b)
Given the linear system 𝑦 𝑛 =2𝑥 3𝑛 , find whether the system is time invariant or not.
Solution:
System
𝑥 1 𝑛
𝑦 1 𝑛 =2 𝑥 1 (3𝑛)
Let the shifted input be:
𝑥 2 𝑛 = 𝑥 1 𝑛− 𝑛 0
Not Equal
Therefore system output:
𝑦 2 𝑛 =2 𝑥 2 3𝑛 =2 𝑥 1 (3𝑛− 𝑛 0 )
Shifting
𝑦 2 𝑛 =2 𝑥 1 (3𝑛)
By 𝑛 0 samples leads to 𝑦 1 𝑛− 𝑛 0 =2 𝑥 1 3 𝑛− 𝑛 0 =2 𝑥 1 3𝑛−3 𝑛 0

20. Difference Equation
A causal, linear, and time invariant system can be represented by a difference equation as follows:
Outputs
Inputs
After rearranging:
Finally:

21. Example 7
Identify non zero system coefficients of the following difference equations.

22. System Representation Using Impulse Response
A linear time-invariant system (LTI system) can be completely described by its unit-impulse response ℎ(𝑛) due to the impulse input 𝛿(𝑛) with zero initial conditions.

23. Example 8 (a)
Consider the difference equation with an initial condition 𝑥 −1 =0.
a. Determine the unit-impulse response ℎ 𝑛 .
b. Draw the system block diagram.
c. Write the output using the obtained impulse response.
Solution

24. Example 8 (b)
Solution

25. Finite Impulse Response (FIR) system:
When the difference equation contains no previous outputs, ( 𝑎 𝑖 coefficients of 𝑦(𝑛−𝑖) are zero).
Infinite Impulse Response (IIR) system:
When the difference equation contains previous outputs, ( 𝑎 𝑖 coefficients of 𝑦(𝑛−𝑖) are not all zero).

26. BIBO Stability BIBO: Bounded In and Bounded Out
A stable system is one for which every bounded input produces a bounded output.
Let, in the worst case, every input value reaches to maximum value M.
Using absolute values of the impulse responses,

27. BIBO Stability
To determine whether a system is stable, we apply the following equation:

28. Example 9 Solution Given a linear system given by:
Which is described by the unit-impulse response:
Determine whether the system is stable or not.
Solution

29. Digital Convolution
A LTI system can be represented using a digital convolution sum. The unit-impulse response ℎ(𝑛) relates the system input and output. To find the output sequence 𝑦(𝑛) for any input sequence 𝑥(𝑛), we use the digital convolution:
The sequences are interchangeable.
Convolution sum requires h(n) to be reversed and shifted.
If h(n) is the given sequence, h(-n) is the reversed sequence.

30. Reversed Sequence Example 10

31. Convolution Using Table Method
Example 11 (a)

32. Convolution Using Table Method Example 11 (b)

33. Convolution Properties
the order in which two signals are convolved makes no difference

34. Examples of Convolution

35. Low Pass Filters
each sample in the output signal being a weighted average of many adjacent points from the input signal.
removing high-frequency
components
(filter's impulse
response)
The rectangular pulse is best at reducing noise while maintaining edge
sharpness.
The Exponential is the simplest recursive filter.
The sinc function is used to separate one band of frequencies from another.

36. High Pass Filters
Typical high-pass filter kernels. These are formed by subtracting the corresponding low-pass filter kernels from a delta function.
The distinguishing characteristic of high-pass filter kernels is a spike surrounded by many adjacent negative samples.

37. Signal-to-Noise Ratio (SNR)
Signal-to-noise ratio ( SNR) is a measure that compares the level of a desired signal to the level of background noise.
 It is defined as the ratio of signal power to the noise power, often expressed in decibels.
If the variance of the signal and noise are known, and the signal is zero-mean:
Because many signals have a very wide dynamic range, SNRs are often expressed using the logarithmic decibel scale. In decibels, the SNR is defined as
A logarithmic scale is a nonlinear scale used when there is a large range of quantities.

38. Signal-to-Noise Ratio (SNR)
The decibel (dB) is a logarithmic unit used to express the ratio between two values of a physical quantity, often power or intensity.

39. Periodicity
Example 12
Consider the following continuous signal for the current
which is sampled at 12.5 ms. Will the resulting discrete signal be periodic?