Classification of signals

Signals are classified into the following categories:

• Continuous Time and Discrete Time Signals
• Deterministic and Non-deterministic Signals
• Even and Odd Signals
• Periodic and Aperiodic Signals
• Energy and Power Signals
• Real and Imaginary Signals

Continuous Time and Discrete Time Signals

A signal is said to be continuous when it is defined for all instants of time.

A signal is said to be discrete when it is defined at only discrete instants of time/

Deterministic and Non-deterministic Signals

A signal is said to be deterministic if there is no uncertainty with respect to its value at any instant of time. Or, signals which can be defined exactly by a mathematical formula are known as deterministic signals.

A signal is said to be non-deterministic if there is uncertainty with respect to its value at some instant of time. Non-deterministic signals are random in nature hence they are called random signals. Random signals cannot be described by a mathematical equation. They are modelled in probabilistic terms.

Even and Odd Signals

A signal is said to be even when it satisfies the condition x(t) = x(-t)

Example 1: t2, t4… cost etc.

Let x(t) = t2
x(-t) = (-t)2 = t2 = x(t)
∴, t2 is even function

Example 2: As shown in the following diagram, rectangle function x(t) = x(-t) so it is also even function.

A signal is said to be odd when it satisfies the condition x(t) = -x(-t)

Example: t, t3 ... And sin t

Let x(t) = sin t
x(-t) = sin(-t) = -sin t = -x(t)
∴, sin t is odd function.

Any function ƒ(t) can be expressed as the sum of its even function ƒ_e(t) and odd function ƒ_o(t).

ƒ(t ) = ƒ_e(t ) + ƒ₀(t )
where
ƒ_e(t ) = ½[ƒ(t ) +ƒ(-t )]

Periodic and Aperiodic Signals

A signal is said to be periodic if it satisfies the condition x(t) = x(t + T) or x(n) = x(n + N).

Where

T = fundamental time period,
1/T = f = fundamental frequency.

The above signal will repeat for every time interval T₀ hence it is periodic with period T₀.

Energy and Power Signals

A signal is said to be energy signal when it has finite energy.

EnergyE=∫∞−∞x2(t)dt

A signal is said to be power signal when it has finite power.

PowerP=limT→∞12T∫T−Tx2(t)dt

NOTE:A signal cannot be both, energy and power simultaneously. Also, a signal may be neither energy nor power signal.

Power of energy signal = 0
Energy of power signal = ∞
Classification of Signals into Power and Energy Signals


Most signals can be classified into Energy signals or Power signals. A signal is classified into an energy or a power signal according to the following criteria


a)         Energy Signals:  an energy signal is a signal with finite energy and zero                                  average power  (0 ≤ E < ¥,  P = 0),
b)         Power Signals:   a power signal is a signal with infinite energy but finite                                 average power  (0 < P < ¥,  E ® ¥).


Comments:


1.         The square root of the average power of a power signal is what is usually defined as the RMS value of that signal.
2.         Your book says that if a signal approaches zero as t approaches ¥ then the signal is an energy signal. This is in most cases true but not always as you can verify in part (d) in the following example.
3.         All periodic signals are power signals (but not all non–periodic signals are energy signals).
4.         Any signal f that has limited amplitude (| f | < ¥) and is time limited
            (f  = 0   for    | t | > t₀  for some  t₀ > 0) is an energy signal as in part (g) in the following example.


Exercise 2–1: determine if the following signals are Energy signals, Power signals, or neither, and evaluate E and P for each signal (see examples 2.1 and 2.2 on pages 17 and 18 of your textbook for help).


            a)         ,




                        This is a periodic signal, so it must be a power signal. Let us prove it.


                        



                       
Notice that the evaluation of the last line in the above equation is infinite because of the first term. The second term has a value between –2 to 2 so it has no effect in the overall value of the energy.


Since a(t)  is periodic with period T = 2p/2p= 1 second, we get


                        





So, the energy of that signal is infinite and its average power is finite (9/2). This means that it is a power signal as expected. Notice that the average power of this signal is as expected (square of the amplitude divided by 2)


            b)         ,





                        Let us first find the total energy of the signal.


                       
                       



The average power of the signal is


                        





So, the signal b(t)  is definitely an energy signal.
So, the energy of that signal is infinite and its average power is finite (9/2). This means that it is a power signal as expected. Notice that the average power of this signal is as expected (the square of the amplitude divided by 2)


 c)    
                ,


d) 


                    
,



                        Let us first find the total energy of the signal.


                        



                       
                        So, this signal is NOT an energy signal. However, it is also NOT a power                            signal since its average power as shown below is zero.
                       
The average power of the signal is


                        





Using Le’hopital’s rule, we see that the power of the signal is zero. That is









So, not all signals that approach zero as time approaches positive and negative infinite is an energy signal. They may not be power signals either.


            e)        

,



            f)        

.





            g)        

.

Real and Imaginary Signals

A signal is said to be real when it satisfies the condition x(t) = x*(t)

A signal is said to be odd when it satisfies the condition x(t) = -x*(t)

Example:

If x(t)= 3 then x*(t)=3*=3 here x(t) is a real signal.
If x(t)= 3j then x*(t)=3j* = -3j = -x(t) hence x(t) is a odd signal.

Note: For a real signal, imaginary part should be zero. Similarly for an imaginary signal, real part should be zero.
REF:www.tutorialspoint.com

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EC6502 Principles of  Digital Signal Processing

Question Bank

UNIT I

Part A

1. Write the formula for DFT and IDFT.
2. Name any two properties of DFT.
3. State and prove Parseval theorem.
4. State and prove time-shifting property of DFT.
5. State and prove circular convolution.
6. What are ‘twiddle factor’ of the DFT?(May-2012)
7. State the relationship between DTFT and DFT.(Nov-2011,May-2014)
8. Determine the value of W¹⁶ for 64-point DFT.
9. Give the number of complex addition and complex multiplication required for the direct computation of N-point DFT.
10. What is zero padding? What are its uses?(Nov-2011,Nov-2013)
11. Define circular convolution.
12. Distinguish between Circular convolution and linear convolution.
13. Write briefly about Overlap-save method.
14. Write briefly about Overlap-add method
15. State the difference between Overlap-save method and Overlap-add method.
16. What is the need for FFT?
17. What is the main advantage of FFT?(May-2011)
18. Calculate the number of complex multiplication and complex addition needed in the calculation of DFT using FFT algorithm with 32-point sequence.(Nov-2010,Nov-2013)
19. What is radix-2 FFT?
20. What is DIT FFT algorithm?
21. What is DIF FFT algorithm?
22. What are the differences and similarities between DIT and DIF algorithm?
23. Draw the basic butterfly diagram of radix 2 DIT FFT.
24. Draw the basic butterfly diagram of radix 2 DIF FFT.
25. What is meant by ‘in-place’ in DIT and DIF algorithm?(Nov-2014)
26. Give the computation efficiency of FFT over DFT.
27. Explain how you would use the FFT algorithm to compute the IDFT.
28. What is bit reversal?(May-2014,May-2011)
29. How many stages of decimations are required in the case of 64pt Radix-2 DIT-FFT algorithm?(May-2012)
30. Given two sequences of length N=4 defined by x[n] = {1, 2, 2, 1} and h[n] = {2, 1, 1, 2}, determine the periodic convolution.

Ans.: y[n] = {9, 10, 9, 8}(Nov-2010)

31. Compute the 4-point DFT of the following sequences,

                                                                i.      x[n] = 2ⁿ

                                                              ii.      x[n] = 2^-n

                                                            iii.      x[n] = sin(nπ/2)

                                                            iv.      x[n] = cos(nπ/2)

Ans.: (a) X(k) = {15, -3+j6, -5,  -3-j6}

(b) X(k) = {15/8, (3/4)-j(3/8), (5/8), (3/4)-j(3/8)}

(c) X(k) = {0, -j2, 0,  j2}

(d) X(k) = {1, 1-j√2, 1, 1+j√2}

32. Find IDFT of X(k) = {1, 0, 1, 0}.

Ans.: x[n] = {0.5, 0, 0.5, 0}

33. Find the IDFT of the sequence X(k) = {10, -2+j2, -2, -2-j2} using DIT and DIF algorithm.

Ans.: x[n] = {1, 2, 3, 4}

34. The first five samples of 8-point DFT of a real valued sequence are {28, -4+j9.565,   -4+j4, -4+j1.656, -4}. Determine the remaining three samples.

Ans.: X(5) = -4-j1.656, X(6) = -4-j4, X(7) = -4-j9.565

35. For the 8-sample sequence x[n] = {1, 2, 3, 5, 5, 3, 2, 1}, the first five DFT coefficients are {22, -7.536-j3.121, 1+j, -0.465-j1.121, 0}. Determine the remaining three DFT coefficients.

Ans.: X(5)= -0.465+j1.121, X(6)= 1-j, X(7)= -7.536+j3.121

36. Consider the finite sequence x[n] = {1, 2, 2, 1}. The 5-point DFT of x[n] is denoted by X(k). Plot the sequence whose DFT is Y(k) = e^-j4πk/5 X(k).

Ans.: y[n] = {1, 0, 1, 2, 2}

37. If the DFT of the sequence x[n] = {1, 2, 1, 1, 2, -1} is X(k). Plot the sequence whose DFT is Y(k) = e^-jπk X(k).

Ans.: y[n] = {1, 2, -1, 1, 2, 1}

38. Consider the 8-point decimation-in-frequency (DIF) flow graph. What is the gain of the “signal path” that goes from x[5] to X(3)?

Ans.: (X(3)/x[5]) = 0.707+j0.707

39. Compute 4-point DFT of a sequence x[n] = {0, 1, 2, 3} using DIF and DIT algorithm.

Ans.: X(k) = {6, -2+j2, -2, -2-j2}

40. Consider the 8-point decimation-in-time (DIT) flow graph. What is the gain of the “signal path” that goes from x[3] to X(2)?

Ans.: (X(2)/x[3]) = j

Part B

1.      An input sequence x[n] = {2, 1, 0, 1, 2} is applied to DSP system having an impulse sequence h[n] = {5,3,2,1}. Determine the output sequence produced by (a) Linear convolution and (b) Verify the same through circular convolution.

Ans.: y[n] = {10, 11, 7, 9, 14, 8, 5, 2}

2.      Convolve the following sequence using (a) Overlap-save method and (b) Overlap-add method,

x[n] = {1, -1, 2, 1, 2, -1, 1, 3, 1} and h[n] = {1, 2, 1}(Nov-2014,Nov-11,Nov-2013,Nov-2010)

Ans.: y[n] = {1, 1, 1, 4, 6, 4, 1, 4, 8, 5, 1}

3.      Draw the signal flow graph for 8/16-point DFT using (a) DIT algorithm and DIF algorithm.(May-2014)

4.       Compute the IDFT for the sequence X(k) = 2^-kwhere k = 0 to 7 using DIF FFT algorithm.(Nov-14)

5.      Find the DFT of a sequence x[n] = {1, 2, 3, 4, 4, 3, 2, 1} using DIT algorithm.(Nov-2010,Nov-2013)

Ans.: X(k) = {20, -5.828-j2.414, 0, 0.172-j0.414, 0, 0.172+j0.414, 0, -5.828-j2.414}  

6.      Find the DFT of a sequence x[n] = {1, 1, 1, 1, 1, 1, 0, 0} using DIF algorithm.  Ans.: X(k) = {6, -0.707-j0.707, 1-j, 0.707+j0.293, 0, 0.707-j0.293, 1+j, -0.707+j0.707}  

7.      Find the DFT of a sequence x[n] = {1, 1, 1, 1, 1, 0, 0, 0} using DIF algorithm.(Nov-2011)

Ans.: X(k) = {5, -j2.414, 1, -j0.414, 1, j0.414, 1,j2.414}  

8.      Find the IDFT of a sequence X(k) = {5, 0, 1-j, 0, 1, 0, 1+j, 0} using DIT algorithm.

Ans.: x[n] = {1, 0.75, 0.5, 0.25, 1, 0.75, 0.5, 0.25}

9.      Consider two sequence x[n] = cos(nπ/2) and h[n] = 2ⁿ. Determine the output sequence y[n] by circular convolution using concentric circle method. Take N=4.

Ans.: y[n] = {-3, -6, 3, 6}

10.  Determine the output sequence y[n] of FIR filter with impulse response, h[n] = {1, 2, 3} to input sequence x[n] = {1, 2, 2, 1}. Use circular convolution in frequency domain.

Ans.: y[n] = {1, 4, 9, 11, 8, 3}

11.  Find the Linear convolution through circular convolution of x₁[n] and x₂[n].

x₁[n] = δ[n] + δ[n-1] + δ[n-2]

x₂[n] = 2δ[n] - δ[n-1] + 2δ[n-2]

Ans.: x₃[n] = 2δ[n] + δ[n-1] + 3δ[n-2] + δ[n-3] + 2δ[n-4]

12.  Given two sequence x₁[n] = {1, 2, 3, 1} and x₂[n] = {4, 3, 2, 2}. Find x3[n] such that (i) X₃(k) = X₁(k).X₂(k), (ii) using concentric circle method and (iii) Matrix method.

Ans.: X₃(k) = {17, 19, 22, 19}

13.  Find the output y[n] of a filter whose impulse response is h[n] = {1, 1, 1} and input signal is x[n] = {3, -1, 0, 1, 3, 2, 0, 1, 2, 1} using (i) Overlap save method and (ii) Overlap add method.

Ans.: y[n] = (3, 2, 2, 0, 4, 6, 5, 3, 3, 4, 3, 1)

14.  Find the output y[n] of a filter whose impulse response is h[n] = {1, 2} and input signal is x[n] = {1, 2, -1, 2, 3, -2, -3, -1, 1, 1, 2, -1} using (i) Overlap save method and (ii) Overlap add method.

Ans.: y[n] = (1, 4, 3, 0, 7, 4, -7, -7, -1, 3, 4, 3, -2)

15.  Determine the output of a linear FIR filter whose impulse response h[n] = {1, -3, 5} and input signal x[n] = {-1, 4, 7, 3, -2, 9, 10, 12, -5, 8} using (i) Overlap-save method and (ii) Overlap-add method.

Ans.: y[n] = (-1, 7, -10, 2, 24, 30, -27, 27, 9, 83, -49, 40)

16.  Determine the DFT of the given data sequence x[n] = {2, 1, 4, 6, 5, 8, 3, 9}(Nov-2013,May-2012)

Ans.: X(k) = {38, -5.828+j6.11, j6, -0.412+j8.1, -10, -0.412-j8.1, -j6, -5.828-j6.11}

17.  Determine the DFT of the given data sequence x[n] = {-1, 2, -3, 4, 9, -20, 12, 6} by DIT and DIF algorithm.

Ans.: X(k) = {9, 6.968+j0.86, -1+j28, -26.968-j29.14, 25, 26.968+j29.14, -1-j28, 6.968-j0.86}

18.  Calculate IDFT for the given coefficients X(k) = {38, -5.828+j6.07, j6, -0.172+j8.07, -10, -0.172-j8.07, -j6, -5.828-j6.07} using DIT and DIF algorithm.

Ans.: x[n] = {2, 1, 4, 6, 5, 8, 3, 9}

19.  State any six properties of DFT(Nov-2014)

20.  Compute IDFT of the sequence X(k) = {7, -0.707-j0.707, -j, 0.707-j0.707, 1, 0.707 +j0.707,  j, -0.707+j0.707} using DIT and DIF algorithm.

Ans.: x[n] = {1, 1, 1, 1, 1, 1, 1, 0}
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