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Frequency response is a critical concept in understanding how a causal linear time-invariant (LTI) system reacts to different frequencies present in an input signal. It's essentially a function describing how each frequency within the input is amplified or attenuated by the system. For any LTI system defined in the frequency domain by \( H(e^{j\omega}) \), the magnitude of \( H(e^{j\omega}) \) represents how the amplitude of that frequency is affected. In our specific system, this function has been designed to maintain a unity magnitude—meaning that for all frequencies \( \omega \), the amplitude does not change. This property is key for filters that are intended to manipulate phase or delay but should not impact the amplitude of the signals passing through them.

The phase angle in the context of frequency response is a measure of how much a sinusoidal input is shifted in time when passed through the system. It's represented mathematically as the angle of \( H(e^{j\omega}) \). In our scenario, the phase angle given by the system's response is calculated to be \( -\omega - 2\tan^{-1}\left(\frac{\frac{1}{2} \sin \omega}{1 - \frac{1}{2} \cos \omega}\right) \). This function describes a frequency-dependent phase shift, with the primary component \( -\omega \) ensuring a basic linear delay consistent at all frequencies. The arctangent term provides additional frequency-specific shifts, adjusting how different frequency components realign as they move through the system.

Group delay, \( \tau(\omega) \), is an important characteristic that describes how the phase of the system varies with frequency. It's the derivative of the phase angle with respect to frequency, often visualized as the time delay experienced by the envelope of a modulated signal. For our LTI system, the group delay is determined by the expression \( \tau(\omega) = \frac{\frac{3}{4}}{\frac{5}{4} - \cos \omega} \). Plotting this, we see the behavior of the group delay changes with \( \omega \), with significant "spikes" occurring in the vicinity of points where the denominator approaches zero but does not reach it due to the bounded range of \( \cos \omega \). These spikes signify areas where the system introduces considerable delay variation as a function of time.

When a cosine input signal like \( \cos(\frac{\pi}{3}n) \) is fed into a causal LTI system, the output can be determined by evaluating the frequency response at the input's frequency components. Cosine wave inputs are split into complex exponentials to get \( x[n] = \frac{1}{2}(e^{j\frac{\pi}{3}n} + e^{-j\frac{\pi}{3}n}) \). Each term responds to the filter according to the system's frequency response: \( H(e^{j\frac{\pi}{3}}) \) and \( H(e^{-j\frac{\pi}{3}}) \). By applying these, the output is a recombination of these components, modified primarily by phase shifts, as the magnitude remains unaffected. The result is a cosine output with the same frequency and amplitude as the input, but potentially phase-shifted due to the behavior captured in the calculated \( \Varangle H(e^{j\omega}) \). This maintains the structure of the wave while adjusting its position in time.