Amplitude Shifting of Continuous-Time Signals

Main ideas / concepts

• The lecture focuses on amplitude shifting of a continuous-time signal.
• It contrasts amplitude scaling vs amplitude shifting:
  • Amplitude scaling multiplies the signal by a factor (e.g., ( \beta \cdot X(t) )), changing the waveform shape (scaling).
  • Amplitude shifting adds a constant (e.g., ( X(t) + K )), which shifts the waveform up or down without changing its basic shape (just a vertical translation).

Signal definitions

• Original signal:

  • ( X(t) ) is defined piecewise as:
    • ( X(t) = 0 ) when ( t < 0 )
    • ( X(t) = 2 ) when ( 0 \le t \le 2 )
    • ( X(t) = 0 ) when ( t > 2 )

• Amplitude-shifted signal:

  • ( Y(t) = X(t) + K )

Method / step-by-step computation (piecewise construction)

For each case, substitute the relevant values of ( X(t) ) into ( Y(t) = X(t) + K ).

Case 1: Upward shifting ((K>0))

• Given: ( K = +2 )
• Then: ( Y(t) = X(t) + 2 )

Compute piecewise:

• If ( t < 0 ): ( Y(t) = 0 + 2 = 2 )
• If ( 0 \le t \le 2 ): ( Y(t) = 2 + 2 = 4 )
• If ( t > 2 ): ( Y(t) = 0 + 2 = 2 )

Interpretation:

• The waveform shifts upward (vertical shift).

Case 2: Downward shifting ((K<0))

• Given: ( K = -2 )
• Then: ( Y(t) = X(t) - 2 )

Compute piecewise:

• If ( t < 0 ): ( Y(t) = 0 - 2 = -2 )
• If ( 0 \le t \le 2 ): ( Y(t) = 2 - 2 = 0 )
• If ( t > 2 ): ( Y(t) = 0 - 2 = -2 )

Interpretation:

• The waveform shifts downward (vertical shift).

Visualization takeaway

• By plotting (X(t)) and (Y(t)):
  • Positive (K) shifts the entire waveform up.
  • Negative (K) shifts the entire waveform down.
• The waveform shape remains the same; only its vertical position changes.

Speakers / sources

• No specific speakers or external sources are named in the subtitles (the content appears to be delivered by the lecturer/instructor).