Frequency Response of Discrete-Time Systems

Understanding the frequency response of a discrete-time system is essential for designing and analyzing digital filters, control systems, and digital signal processors. It reveals how a system modifies the amplitude and phase of sinusoidal inputs, dictating performance in applications from audio processing to telecommunications. This analysis bridges the gap between abstract system models and real-world behavior, with unique constraints like periodicity and aliasing that don't exist in the continuous-time world.

From Z-Transform to Frequency Response

The z-transform is the fundamental tool for analyzing linear time-invariant (LTI) discrete-time systems. For a system with an impulse response $h[n]$, the z-transform is defined as $H(z)={\sum }_{n=-\infty }^{\infty }h[n]z^{-n}$, where $z$ is a complex variable. This transfer function $H(z)$ completely characterizes the system's input-output relationship in the z-domain.

The frequency response is obtained by evaluating this z-domain transfer function on the unit circle in the complex plane. Mathematically, we substitute $z=e^{j\omega T}$, where $j$ is the imaginary unit, $\omega$ is the continuous-time angular frequency in radians per second, and $T$ is the sampling period. The resulting function, $H(e^{j\omega T})$, is the frequency response. It is a complex-valued function of $\omega$, whose magnitude $|H(e^{j\omega T})|$ represents the system's gain (or attenuation) and whose argument $\angle H(e^{j\omega T})$ represents the phase shift imparted to a sinusoidal input at frequency $\omega$.

For example, consider a simple moving average filter with difference equation $y[n]=0.5(x[n]+x[n-1])$. Its transfer function is $H(z)=0.5(1+z^{-1})$. To find the frequency response, we evaluate on the unit circle: $H(e^{j\omega T})=0.5(1+e^{-j\omega T})$. Using Euler's formula, this can be manipulated to show the magnitude response is $|\cos (\omega T/2)|$, confirming it attenuates higher frequencies.

Periodicity and the Sampling Frequency

A defining property of the discrete-time frequency response $H(e^{j\omega T})$ is that it is periodic in $\omega$. The period is equal to the sampling frequency ${\omega }_s=2\pi /T$ radians per second (or $f_s=1/T$ Hz). This means $H(e^{j(\omega +{\omega }_s)T})=H(e^{j\omega T})$ for all $\omega$.

This periodicity arises directly from the substitution $z=e^{j\omega T}$. The complex exponential $e^{j\omega T}$ is periodic in $\omega$ with period ${\omega }_s$. Since the frequency response is only evaluated at these periodic points on the unit circle, the function itself must repeat. Consequently, we typically only need to examine one period, most commonly the baseband or fundamental frequency range from $-{\omega }_s/2$ to ${\omega }_s/2$ (or $0$ to ${\omega }_s/2$ for the magnitude response due to symmetry for real-valued systems).

This periodicity has a profound practical implication: a discrete-time system responds identically to input sinusoids at frequencies $\omega$ and $\omega +k{\omega }_s$, where $k$ is any integer. In a digital audio system with a sampling rate $f_s=44.1$ kHz, a filter's effect on a 1 kHz tone is identical to its effect on a 45.1 kHz tone—a frequency far outside the range of human hearing.

Aliasing and Bandwidth Limitation

The phenomenon of aliasing is inextricably linked to the periodicity of the discrete-time frequency response. Aliasing occurs when a sinusoid with a frequency outside the baseband range is sampled. The resulting discrete-time sequence is indistinguishable from a sequence generated by sampling a lower-frequency sinusoid within the baseband.

This "folding" of high-frequency content into the baseband is not just a property of the sampling process but fundamentally shapes system bandwidth. Because the frequency response is periodic, any discrete-time system treats an aliased frequency component as if it were its baseband counterpart. Therefore, the effective bandwidth of any discrete-time system is fundamentally limited to frequencies within $\pm f_s/2$ Hz. No discrete-time LTI system can independently process or distinguish between two input frequencies separated by an integer multiple of the sampling frequency.

This forces a critical design constraint: to avoid aliasing of unwanted input signals, an anti-aliasing filter (a continuous-time low-pass filter) must be used before the analog-to-digital converter to attenuate all frequency components above $f_s/2$. The discrete-time system then operates only on the correctly represented baseband frequencies.

Common Pitfalls

1. Ignoring Aliasing in System Analysis: A common mistake is to analyze a discrete-time system's response to a high-frequency input without considering aliasing. For example, calculating the output for an input at $1.2f_s$ by directly plugging that frequency into $H(e^{j\omega T})$ is misleading. The correct approach is to first find the aliased frequency within the baseband. Since $1.2f_s=f_s+0.2f_s$, it will alias down to $0.2f_s$. The system's output is determined by its frequency response at $0.2f_s$, not at $1.2f_s$.

1. Confusing Continuous and Discrete Frequency: Engineers sometimes misinterpret the period ${\omega }_s$ on a frequency response plot. Remember, $\omega$ on the horizontal axis is the continuous-time input frequency. The response repeats every ${\omega }_s$ (e.g., $2\pi /T$), not every $2\pi$. The normalized frequency $\hat{\omega }=\omega T$, which ranges from $-\pi$ to $\pi$, is often used to make the period $2\pi$, but the underlying physical period is still ${\omega }_s$.

1. Overlooking Phase Response Periodicity: While the magnitude response $|H(e^{j\omega T})|$ is periodic and even-symmetric for real systems, the phase response $\angle H(e^{j\omega T})$ is periodic and odd-symmetric. Forgetting this can lead to errors in applications sensitive to phase, such as audio crossovers or control loops, where phase wrapping (jumps of $2\pi$) must be handled correctly to interpret the true phase lag.

Summary

• The frequency response of a discrete-time LTI system is found by evaluating its z-domain transfer function $H(z)$ on the unit circle, setting $z=e^{j\omega T}$.
• The response $H(e^{j\omega T})$ is periodic in the continuous-time frequency $\omega$, with a period equal to the sampling frequency ${\omega }_s=2\pi /T$. This confines meaningful analysis typically to the baseband range $-{\omega }_s/2\le \omega \le {\omega }_s/2$.
• Aliasing is a direct consequence of this periodicity, causing any frequency outside the baseband to be irretrievably folded into it. This fundamentally limits a discrete-time system's operational bandwidth to less than half the sampling rate, mandating the use of anti-aliasing filters in practice.