LTI Systems: Frequency Domain Analysis

Understanding how Linear Time-Invariant (LTI) systems behave when subjected to various input signals is fundamental in signal processing. LTI Systems: Frequency Domain Analysis provides a powerful framework for this, particularly when dealing with sinusoidal inputs. Instead of analyzing complex convolutions in the time domain, the frequency domain offers a simpler and more intuitive approach, revealing how an LTI system alters the amplitude and phase of different frequency components.

What is LTI Systems Frequency Domain Analysis?

LTI systems are characterized by their impulse response, g[n]. When an analytic (complex-valued) sinusoidal signal, x[n] = A * e^(jΩn), is applied as input to an LTI system, the output y[n] is also an analytic sinusoid of the same frequency. This crucial property simplifies analysis significantly.

The relationship between the input and output is defined by a complex constant known as the frequency response, denoted as K(Ω). The output signal can be expressed as: y[n] = K(Ω) * A * e^(jΩn).

This frequency response, K(Ω), is equivalent to the system's transfer function G(z) evaluated on the unit circle in the z-plane, specifically K(Ω) = G(e^(jΩ)). It tells us two key things:

• The magnitude, |K(Ω)|, which scales the amplitude of the input sinusoid.
• The phase, arg(K(Ω)), which shifts the initial phase of the input sinusoid.

Generalized Frequency Response in LTI Systems

When the input signal is a complex sinusoid, x[n] = A * e^(jΩn), the output signal y[n] for an LTI system g[n] can be found by leveraging the system's linearity. The relationship remains the same as for a normalized analytic signal where amplitude A=1.

The output signal is given by y[n] = G(e^(jΩ)) * A * e^(jΩn), where G(e^(jΩ)) is the frequency response K(Ω). This means the system essentially multiplies the input by a frequency-dependent complex constant.

Analytic Signals: A Practical Tool

While real-valued sinusoidal signals are what naturally exist, their complex variant, often called an analytic sinusoidal signal, s_hat(t) = A * e^(j(ωt + φ)), is more practical for LTI system analysis. In discrete time, this becomes s_hat[n] = A * e^(j(Ωn + φ)).

An analytic signal is difficult to plot directly. We typically visualize it using:

• Cartesian coordinates: Two separate plots for the real and imaginary parts.
• Polar coordinates: A trajectory in the complex plane, representing a rotating phasor.

Processing Real-Valued Sinusoids

For an LTI system with a real-valued sinusoid input, x[n] = cos(Ωn), the output y[n] can be calculated using the properties of the frequency response. The output will also be a real-valued sinusoid with a modified amplitude and phase shift:

y[n] = |K(Ω)| * cos(Ωn + arg(K(Ω)))

Here, |K(Ω)| is the magnitude of the frequency response and arg(K(Ω)) is its phase, where K(Ω) = G(e^(jΩ)) = |K(Ω)| * e^(j * arg(K(Ω))).

Key Properties of Frequency Response

The frequency response K(Ω) exhibits several important properties:

Periodicity

The function K(Ω) is periodic with a period of 2π. This means that K(Ω) = K(Ω + 2πk) for any integer k. This property arises because e^(j2πk) = 1.

K(Ω + 2πk) = G(e^(j(Ω + 2πk))) = G(e^(jΩ) * e^(j2πk)) = G(e^(jΩ) * 1) = G(e^(jΩ)) = K(Ω).

Symmetry

For an LTI system with a real-valued impulse response g[n], the frequency response K(Ω) exhibits conjugate symmetry: K*(-Ω) = K(Ω). This implies that |K(-Ω)| = |K(Ω)| and arg(K(-Ω)) = -arg(K(Ω)).

This property is derived from the definition of the frequency response K(Ω) = Σ g[n] * e^(-jΩn) and the fact that g[n] is real-valued.

Plotting the Frequency Response

To visualize an LTI system's behavior across different frequencies, it is common to plot two graphs of its frequency response as a function of frequency Ω:

1. Magnitude Response: |K(Ω)|, showing how the system amplifies or attenuates different frequencies.
2. Phase Response: arg(K(Ω)), showing how the system shifts the phase of different frequencies.

In MATLAB or Python, the freqz() function is used for this purpose. It calculates the frequency response for a given set of numerator B and denominator A coefficients of the system's transfer function.

Example: System G(z) = 0.2 / (1 - 0.8z^-1)

Let's consider the system with the transfer function G(z) = 0.2 / (1 - 0.8z^-1). If the input is x[n] = e^(j(2π/40)n), we can calculate the output:

1. Frequency Response: Evaluate G(z) at z = e^(j(2π/40)). The frequency response K(Ω) at Ω = 2π/40 is approximately 0.8186 * e^(-j0.5377). This means the amplitude is scaled by 0.8186 and the phase is shifted by -0.5377 radians.
2. Output Signal: y[n] = 0.8186 * e^(j((2π/40)n - 0.5377)).

When comparing frequency domain analysis with time domain calculation (e.g., using filter() or lfilter()), a transient region may appear in the time domain. This is because time domain computation often implicitly assumes an input starting at n=0, effectively skipping the n ∈ (-∞, -1> part of the signal. Frequency domain analysis inherently assumes steady-state conditions, similar to