Final answer:
To express the given signal in the frequency domain, we use Fourier analysis. the frequency domain representation of the given signal is Y(w) = 2π[δ(w−4π×10³) − δ(w+4π×10³)] + 2π[δ(w−10π×10³) − δ(w+10π×10³)]. sketching the time and frequency waveforms will depend on the time interval and frequency range considered.
Step-by-step explanation:
The given expression in the time domain is y(t) = 2 sin(2π × 2 × 10³ t) + 2 sin(2π × 5 × 10³ t). to express this signal in the frequency domain, we use fourier analysis. fourier analysis allows us to decompose a signal into its constituent frequencies. the general form of a Fourier transform is given by F(w) = ∫[f(t)e^(−jwt)]dt, where F(w) is the frequency domain representation, f(t) is the time domain representation, and ∫ denotes integration.
To solve the problem, we need to know that the fourier transform of sin(at) is π[δ(w−a) − δ(w+a)] and the Fourier transform of cos(at) is π[δ(w−a) + δ(w+a)]. using this information, the frequency domain representation of the given signal is Y(w) = 2π[δ(w−4π×10³) − δ(w+4π×10³)] + 2π[δ(w−10π×10³) − δ(w+10π×10³)].
To sketch the time and frequency waveforms, we can use the information from the frequency domain representation. In the time domain, the waveform will have two sinusoidal components at frequencies 4π×10³ and 10π×10³, each with an amplitude of 2. in the frequency domain, we will have two peaks at frequencies 4π×10³ and 10π×10³, each with an amplitude of 2π. The shape of the waveforms will depend on the time interval and frequency range considered.