1. Basic Continuous-Time Signals
Unit step function:
u t = 0, t<0 1, t>0
u(t) is discontinuous at t=0, and its value at t=0 is not defined.
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

2. r t = 0, 0< t<T 1, t<0 and t>T
Rectengular pulse:
r t = 0, < t<T 1, t<0 and t>T
Note that rectangular pulse can be expressed in terms of unit-step function; r(t)=u(t)-u(t-T)
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

3. Unit Impulse Function:
Unit impulse function is also called as ‘Dirac delta’ function
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

4. t 1 t 2 δ t dt = 1, if t 1 <0< t 2 0, otherwise
δ t = for t≠0
δ t is undefined for t=0
t 1 t 2 δ t dt = 1, if t 1 <0< t 2 0, otherwise
Unit impulse (dirac delta) is the derivative of the unit step;
δ t = du(t) dt ⟹u(t) −∞ t δ(τ)dτ
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

5. Exponential Signals x t =C e at What about if a=0 ?
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

6. Basic Modification of Continuous-Time Signals
Shifting of a signal:
y(t)=x(t-t0)
if t 0 >0, it is called ′delay′
if t 0 <0, it is called ′advance′
Example: Let x(t) be unit step function, u(t). If y(t) is defined as y(t)=u(t-5), we can sketch y(t) as
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

7. r t = 0, 2< t<5 1, otherwise
Example: Define rectangular function as
r t = 0, < t<5 1, otherwise
Sketch r(t).
Sketch the shifted signal, z(t)=r(t+3)
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

8. Axis reversal of signal:
Axis reversed version of any continuous-time x(t) is defined as y t =x(−t)
Example: If x t =A e αt α>0 , sketch the axis reversed signal y t =x −t =A e αt
Sketch the shifted signal, z(t)=r(t+3)
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

9. Sketch the shifted signal, z(t)=r(t+3) Scaling of a signal:
Scaled version of a continuous-time signal x(t) is defined as y t =x at , where a>0.
if a>1, y t is a ′ c ompressed ′ version of x t .
if 0<a<1, y t is a ′ strec hed′ version of x t .
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

10. Discrete-time real exponential can be defined as
Example:
Exponential signals
Discrete-time real exponential can be defined as
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

11. x n =C a n , C and a are real numbers.
As can be seen in 4 different cases above, exponential signals grow with n for a >1 and decay with n for a <1.
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

12. Also, signs of the samples alternate with n if a<0.
What about for a=1 and a=-1 ?
Basic Modifications of Discrete*Time Signals:
Shifting of a signal:
y n =x[n− n 0 ]
if n 0 >0, it is called ′delay′
if n 0 <0, it is called ′advance′
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

13. Example:
Let x[n] br the ‘unit impulse’ signal δ[n], and let n0=-3. Then, y[n] is given as
y[n]=δ[n-(-3)]=δ[n+3]
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

14. Axis reversal of a signal:
Axis reversed version of any discrete-time signal x[n] is defined by y n =x[−n]
Example:
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

15. Scaling of a signal: y n =x[a.n]
If we let y[n]=x[2n], this coressponds to compression of the signal x[n] by a factor of 2. Note that since y[0]=x[0], y[1]=x[2], y[2]=x[4], etc., y[n] contains only the even-numbered samples of x[n]. This is called decimation of x[n].
To obtain only the odd-numbered samples of x[n], we could let w[n]=x[2n+1] (w[0]=x[1], w[1]=x[3], w[2]=x[5], etc.)
What happens if a<1 ?
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

16. 𝑧 𝑛 = x n 2 ; n even x (n−1) 2 +x[ (n−1) 2 ] 2 ; n odd
There are some difficulties. For example, z n =x[ n 2 ] is not defined for odd values of n. (x[ 1 2 ], x[ 3 2 ],... do not exist)
Hence, in order to stretch a discrete-time signal we must interpolate between given values of x[n]. For example, if we use simple linear interpolation, we could define z[n] as
𝑧 𝑛 = x n 2 ; n even x (n−1) 2 +x[ (n−1) 2 ] 2 ; n odd
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

17. Example of interpolation&decimation:
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

18. Combinations of shifting, scaling&axis-reversal operations:
Example: x(t) is given as the ‘trapezoidal’ signal. Find and draw the modified signal y(t)=x(1+t/2).
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

19. Solution 1: ‘advance by 1’ ‘stretch by 2’ x t ⟶x t+1 ⟶x t 2 +1
Solution 2: ‘stretch by 1’ ‘advance by 2’
x t ⟶x t 2 ⟶x t+2 2 =x( t 2 +1)
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

20. y(-1)=x(1+(-1)/2)=x(1-0.5)=x(0.5)=1 You can try even more points.
Check the answer:
y(2)=x(1+2/2)=x(1+1)=x(2)=0
y(0)=x(1+0/2)=x(1)=1
y(-1)=x(1+(-1)/2)=x(1-0.5)=x(0.5)=1
You can try even more points.
Example: Let y[n]=u[2-n] (shifted and axis-reversed discrete-time unit step). Draw y[n].
Solution 1: ‘axis reversal’ ‘delay by 2’
u n ⟶u −n ⟶u − n−2 =u −n+2 =y[n]
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

21. Solution 2: ‘advanced by 2’ ‘axis-reversal’ u n ⟶u n+2 ⟶u −n+2 =y[n]
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

22. Example: x[n] is given as
Check the answer:
y[2]=u[-2+2]=u[0]=1
y[-2]=u[-(-2)+2]=u[4]=1
y[4]=u[-4+2]=u[-2]=0
Example: x[n] is given as
Draw the modified signal y[n]=x[2-2n].
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

23. ‘decimate by 2’ ‘axis-reversal’ ‘delay by 1’
x n ⟶𝑥 2n ⟶𝑥 −2n ⟶x −2 n−1
=x −2n+2
=y n
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

24. Amplitude Scaling of Signals
Continuous-time:
y(t)=Ax(t); ∀ t∈ℝ, A is a constant A∈ℝ
A x(t)→y(t) Block Diagram Representation
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

25. y[n]=Ax[n]; ∀ n∈ℤ, A is a constant A∈ℝ
Discrete-time:
y[n]=Ax[n]; ∀ n∈ℤ, A is a constant A∈ℝ
A x[n]→y[n] Block Diagram Representation
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

26. Multiplication (Product) of Signals
Continuous-time:
y(t)=x1(t)x2(t); ∀ t∈ℝ
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

27. Discrete-time: y[n]=x1[n]x2[n]; ∀ n∈ℤ
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

28. Example: x1(t)=t, x2(t)=u(t) y(t)=x1(t)x2(t)=tu(t)
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29. Similarly, a discrete-time ramp signal can be defined as
y n = n; n≥0 0; n<0
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

30. Example: x1[n]=2n-1, x2[n]=-δ[n+3]
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

31. Addition of Signals Continuous-time: y(t)=x1(t)+x2(t); ∀ t∈ℝ
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

32. x 1 t = t; t≥0 0; t<0 cts−time (ramp signal)
Example:
x 1 t = t; t≥0 0; t<0 cts−time (ramp signal)
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

33. Discrete-time: y[n]=x1[n]+x2[n]; ∀ n∈ℤ
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

34. Define rectangular pulse p n = 0, n<0 1, 0≤n≤2 0, n≥3
Example:
Define rectangular pulse
p n = 0, n<0 1, 0≤n≤2 0, n≥3
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING