# confused in using modulation technique

I am given a question in which I have to draw the spectrum of the signal $$m(t)$$ and then draw the spectrum of this modulated signal. The signal is

$$m(t)=\cos(1000\pi t)$$

Now in solution manual what they have done is that they have staright away drawn two pulses. One pulse in positive x-axis at $$1000$$ and at at negative x-axis at$$-1000$$. I was looking at the online solution of this question and in that solution following steps were taken:

We know that the Fourier Transform pair is: \begin{align} \cos(2\pi f_ot) &\leftrightarrow \frac{1}{2}[\delta(f-f_o)+\delta(f+f_o)]\\ m(t)&= \cos(1000\pi t)\\ &= \cos(2\pi 500t) \end{align}

Apply Fourier Transform and we get: $$M_1(f)= \frac{1}{2}[\delta(f-500)+\delta(f+500)]$$ Hence two pulses of are drawn at 500 and -500. Which of the above mentioned method is correct. The solution manual is of Modern and Analog Digital Communication 4th edition.

• you're right, the solution manual isn't. – Marcus Müller Apr 19 at 15:15