Ct signal operations
- 1. Operations on Continuous-Time SignalsContinuous-Time Signals David W. Graham EE 327
- 2. Continuous-Time Signals • Continuous-Time Signals – Time is a continuous variable – The signal itself need not be continuous 2 • We will look at several common continuous-time signals and also operations that may be performed on them
- 3. Unit Step Function u(t) ( ) < ≥ = 00 01 t t tu t 1 3 •Used to characterize systems •We will use u(t) to illustrate the properties of continuous-time signals t 0
- 4. Operations of CT Signals 1. Time Reversal y(t) = x(-t) 2. Time Shifting y(t) = x(t-td) 3. Amplitude Scaling y(t) = Bx(t) 4 4. Addition y(t) = x1(t) + x2(t) 5. Multiplication y(t) = x1(t)x2(t) 6. Time Scaling y(t) = x(at)
- 5. 1. Time Reversal • Flips the signal about the y axis • y(t) = x(-t) ex. Let x(t) = u(t), and perform time reversal 5 Solution: Find y(t) = u(-t) Let “a” be the argument of the step function u(a) ( ) < ≥ = 00 01 a a au Let a = -t, and plug in this value of “a” ( ) > ≤ =− 00 01 t t tu t 1 0
- 6. 2. Time Shifting / Delay • y(t) = x(t – td) • Shifts the signal left or right • Shifts the origin of the signal to td 6 • Rule Set (t – td) = 0 (set the argument equal to zero) Then move the origin of x(t) to td • Effectively, y(t) equals what x(t) was td seconds ago
- 7. 2. Time Shifting / Delay ex. Sketch y(t) = u(t – 2) Method 1 Let “a” be the argument of “u” ( ) ≥ = ≥− = ≥ = 2102101 tta ay 7 ( ) < = <− = < = 2002000 tta ay t 1 0 1 2 Method 2 (by inspection) Simply shift the origin to td = 2
- 8. 3. Amplitude Scaling • Multiply the entire signal by a constant value • y(t) = Bx(t) ex. Sketch y(t) = 5u(t) 8 t 5 0
- 9. 4. Addition of Signals • Point-by-point addition of multiple signals • Move from left to right (or vice versa), and add the value of each signal together to achieve the final signal • y(t) = x (t) + x (t) 9 • y(t) = x1(t) + x2(t) • Graphical solution – Plot each individual portion of the signal (break into parts) – Add the signals point by point
- 10. 4. Addition of Signals ex. Sketch y(t) = u(t) – u(t – 2) First, plot each of the portions of this signal separately •x1(t) = u(t) Simply a step signal •x2(t) = –u(t-2) Delayed step signal, multiplied by -1 10 t 1 1 2 -1 x1(t) = u(t) x2(t) = -u(t - 2) Then, move from one side to the other, and add their instantaneous values t 1 0 1 2 y(t)
- 11. 5. Multiplication of Signals • Point-by-point multiplication of the values of each signal • y(t) = x1(t)x2(t) 11 • Graphical solution – Plot each individual portion of the signal (break into parts) – Multiply the signals point by point
- 12. 5. Multiplication of Signals ex. Sketch y(t) = u(t)·u(t – 2) First, plot each of the portions of this signal separately •x1(t) = u(t) Simply a step signal •x2(t) = u(t-2) Delayed step signal 12 t 1 1 2 x1(t) = u(t) x2(t) = -u(t - 2) Then, move from one side to the other, and multiply instantaneous values t 1 1 2 y(t)
- 13. 6. Time Scaling • Speed up or slow down a signal • Multiply the time in the argument by a constant • y(t) = x(at) 13 • y(t) = x(at) |a| > 1 Speed up x(t) by a factor of “a” |a| < 1 Slow down x(t) by a factor of “a” • Key Replace all instances of “t” with “at”
- 14. 6. Time Scaling ex. Let x(t) = u(t) – u(t – 2) Sketch y(t) = x(2t) 1 x(t) First, plot x(t) Replace all t’s with 2t 14 t1 2 y(t) = x(2t) = u(2t) – u(2t – 2) Turns on at 2t ≥ 0 t ≥ 0 No change Turns on at 2t - 2 ≥ 0 t ≥ 1 t 1 1 2 y(t) This has effectively “sped up” x(t) by a factor of 2 (What occurred at t=2 now occurs at t=2/2=1)
- 15. 6. Time Scaling ex. Let x(t) = u(t) – u(t – 2) Sketch y(t) = x(t/2) 1 x(t) First, plot x(t) Replace all t’s with t/2 15 t1 2 y(t) = x(t/2) = u(t/2) – u((t/2) – 2) Turns on at t/2 ≥ 0 t ≥ 0 No change Turns on at t/2 - 2 ≥ 0 t ≥ 4 This has effectively “slowed down” x(t) by a factor of 2 (What occurred at t=1 now occurs at t=2) t 1 1 2 y(t) 3 4
- 16. Combinations of Operations • Combinations of operations on signals – Easier to Determine the final signal in stages – Create intermediary signals in which one operation is performed 16 operation is performed
- 17. ex. Time Scale and Time Shift Let x(t) = u(t + 2) – u(t – 4) Sketch y(t) = x(2t – 2) t 1 1 2 x(t) 3 4-1-2 Can perform either operation first Method 1 Shift then scale 17 Method 1 Shift then scale Let v(t) = x(t – b) Time shifted version of x(t) Then y(t) = v(at) = x(at – b) Replace “t” with the argument of “v” Match up “a” and “b” to what is given in the problem statement at – b = 2t – 2 (Match powers of t) a = 2 b = 2 Therefore, shift by 2, then scale by 2 t 1 1 2 v(t) = x(t - 2) 3 4 5 6 t 1 1 2 y(t) = v(2t) = x(2t - 2) 3 4 5 6
- 18. ex. Time Scale and Time Shift Let x(t) = u(t + 2) – u(t – 4) Sketch y(t) = x(2t – 2) t 1 1 2 x(t) 3 4-1-2 Can perform either operation first Method 2 Scale then shift 18 Method 2 Scale then shift Let v(t) = x(at) Time scaled version of x(t) Then y(t) = v(t – b) = x(a(t – b)) = x(at – ab) Replace “t” with the argument of “v” Match up “a” and “b” to what is given in the problem statement at – ab = 2t – 2 (Match powers of t) a = 2 ab = 2, b = 1 Therefore, scale by 2, then shift by 1 t 1 1 2 v(t) = x(2t) 3 4-1-2 t 1 1 2 y(t) = v(t - 1) = x(2(t - 1)) x(2t - 2) 3 4-1-2
- 19. ex. Time Scale and Time Shift • Note – The results are the same • Note – The value of b in Method 2 is a scaled version of the time delay – t = 2 19 – td = 2 – Time scale factor = 2 – New scale factor = 2/2 = 1