To sketch the voltage spectrum for a 1 MHz carrier modulated by a 1 kHz signal with a modulation index of $m = 0.5$, we need to break down the components of the frequency spectrum.

Steps for Understanding:

1. Carrier Signal:

   • The carrier frequency $f_c$ is 1 MHz (1,000,000 Hz).
   • The peak amplitude of the carrier is 1 V.

2. Modulation Signal:

   • The modulating signal frequency $f_m$ is 1 kHz.
   • The modulation index $m$ is 0.5.

3. Frequency Spectrum of Amplitude Modulated (AM) Signal: For a standard AM signal, the voltage spectrum consists of:

   • Carrier Frequency: The unmodulated signal at $f_c$.
   • Sidebands: These are located at frequencies $f_c - f_m$ and $f_c + f_m$, and their amplitudes are determined by the modulation index $m$.

   In general, the voltage spectrum of an AM signal consists of:

   • A carrier at frequency $f_c$.
   • Two sidebands at $f_c + f_m$ and $f_c - f_m$.

   The sideband frequencies can be calculated as:

   • $f_c + f_m = 1,000,000 + 1,000 = 1,001,000$ Hz
   • $f_c - f_m = 1,000,000 - 1,000 = 999,000$ Hz

4. Sideband Amplitudes:

   • The amplitudes of the sidebands for AM modulation are proportional to the carrier amplitude and the modulation index $m$.
   • The amplitude of each sideband is given by: $A_{\text{sideband}} = \frac{m}{2} \cdot A_{\text{carrier}}$
     • For $m = 0.5$ and $A_{\text{carrier}} = 1 \, \text{V}$, the amplitude of each sideband is: $A_{\text{sideband}} = \frac{0.5}{2} \cdot 1 = 0.25 \, \text{V}$

5. Resulting Spectrum:

   • Carrier at 1 MHz, with an amplitude of 1 V.
   • Sideband 1 at 999 kHz, with an amplitude of 0.25 V.
   • Sideband 2 at 1.001 MHz, with an amplitude of 0.25 V.

Voltage Spectrum Sketch:

1. Place a spike at 1 MHz with a height of 1 V (the carrier).
2. Place a smaller spike at 999 kHz with a height of 0.25 V (first sideband).
3. Place a similar smaller spike at 1.001 MHz with a height of 0.25 V (second sideband).

The spectrum consists of these three spikes with the following frequencies and amplitudes:

• 1 MHz: 1 V (Carrier)
• 999 kHz: 0.25 V (Lower Sideband)
• 1.001 MHz: 0.25 V (Upper Sideband)

Would you like a more detailed explanation or a diagram of this spectrum?

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Related Questions:

1. How does the modulation index $m$ affect the bandwidth of an AM signal?
2. What is the mathematical formula for the bandwidth of an AM signal?
3. How would the spectrum change if the modulation index were 1 instead of 0.5?
4. What happens to the sidebands if the modulating frequency is increased further?
5. How can we use a Fourier transform to calculate the exact spectrum of an AM signal?

Tip: The total bandwidth of an AM signal is typically $2 \cdot f_m$, meaning the distance between the two sidebands is $2 \cdot f_m$.