Define the frequency response H(jω) of an LTI system and describe its relationship to the impulse response.

• A. H(jω) = ∫ h(t) e^{−jω t} dt; Y(jω) = X(jω) H(jω); The impulse response and transfer function are Fourier transform pairs.
• B. H(jω) = ∑ h[n] e^{−jω n}
• C. H(jω) = ∫ x(t) e^{−jω t} dt.
• D. H(jω) is the derivative of h(t).

Answer: (A)

The frequency response of an LTI system is found by taking the Fourier transform of its impulse response. This impulse response h(t) is the system’s output when the input is a delta function, so it completely characterizes the system in time. Translating to the frequency domain, H(jω) = ∫_{-∞}^{∞} h(t) e^{−jωt} dt. This complex function tells you, for each frequency ω, how that sinusoidal input is scaled and shifted in phase: the magnitude |H(jω)| shows amplification or attenuation, and the angle ∠H(jω) gives the phase shift.

For any input x(t) with Fourier transform X(jω), the output in the frequency domain is Y(jω) = X(jω)H(jω); in time, y(t) is the convolution x(t) * h(t). The impulse response and transfer function are Fourier transform pairs, so knowing h(t) is equivalent to knowing H(jω).

The other options mix up what is being transformed or describe an incorrect relation—for example, transforming the input instead of the impulse response, or stating a derivative relationship that doesn’t define the transfer function.