4.5.1 Concept of Transfer Function (Frequency Response, H(jω)):

Recall LTI System Output: We know from earlier modules that the output y(t) of a CT-LTI system with impulse response h(t), subjected to an input x(t), is given by the convolution integral:

y(t) = x(t) * h(t) (where '*' denotes convolution)
y(t) = Integral from tau = -infinity to +infinity of (x(tau) * h(t - tau) d(tau))

The Power of the Convolution Property: As established in section 4.3.7, the Fourier Transform of a convolution in the time domain is a simple multiplication in the frequency domain. Applying the FT to the convolution equation:

F{y(t)} = F{x(t) * h(t)}
Y(jω) = X(jω) * H(jω)

Definition of Transfer Function / Frequency Response:
1. H(jω) is defined as the Fourier Transform of the system's impulse response h(t):
H(jω) = F{h(t)} = Integral from t = -infinity to +infinity of (h(t) * e^(-jωt) dt)
2. Physical Meaning: The term "Frequency Response" is highly descriptive. It literally tells you how the system responds to different frequencies. If you input a complex exponential e^(jωt) (a single frequency), the output of an LTI system will be H(jω) * e^(jωt). So, H(jω) is the complex scaling factor (gain and phase shift) that the system applies to that particular frequency component.

Significance:
1. Complete Characterization: Just as the impulse response h(t) completely characterizes an LTI system in the time domain, its Fourier Transform, H(jω), completely characterizes the same LTI system in the frequency domain. They are two sides of the same coin.
2. Simplified Analysis: The most profound simplification is that a complex time-domain convolution becomes a straightforward multiplication in the frequency domain. This allows for very intuitive analysis of filter design, modulation, and other signal processing tasks.

4.5.2 Magnitude and Phase Spectra of H(jω):

Since H(jω) is generally a complex-valued function of frequency, it can be uniquely represented by its magnitude and phase at each frequency.
H(jω) = |H(jω)| * e^(j * angle(H(jω)))

Magnitude Response (|H(jω)|):
1. Interpretation: This component of the frequency response tells us the gain of the system at each specific frequency.

• If |H(jω)| > 1: The system amplifies that frequency component.
• If |H(jω)| < 1: The system attenuates (reduces the amplitude of) that frequency component.
• If |H(jω)| = 1: The system passes that frequency component without changing its amplitude.
• Plotting: Typically plotted on a logarithmic scale (decibels, dB) against frequency. For example, 20 * log10(|H(jω)|) dB.

Phase Response (angle(H(jω))):
1. Interpretation: This component tells us the phase shift (or delay/advance) that the system imparts to each specific frequency component.
- A non-linear phase response can lead to phase distortion (or group delay distortion), where different frequency components experience different amounts of delay, causing the shape of the signal to change.
- An ideal phase response for many applications is linear phase, meaning angle(H(jω)) = -k * ω (for some positive constant k). A linear phase response corresponds to a pure time delay of 'k' seconds for all frequency components, which means the signal's shape is preserved (just shifted in time).

1. Plotting: Typically plotted in radians or degrees against frequency.

Combined Effect on Output Spectrum:
Y(jω) = X(jω) * H(jω)
Y(jω) = (|X(jω)| * |H(jω)|) * e^(j * (angle(X(jω)) + angle(H(jω))))
1. The magnitude of the output spectrum at any frequency is the product of the input magnitude and the system's magnitude response at that frequency.
2. The phase of the output spectrum at any frequency is the sum of the input phase and the system's phase response at that frequency.

4.5.3 Ideal Filters: Low-pass, High-pass, Band-pass, Band-stop:

Concept: Filters are systems specifically designed to selectively alter the frequency content of a signal. They are fundamental components in nearly all signal processing and communication systems. Ideal filters are theoretical constructs that represent the perfect, desired behavior for a filter. While not physically realizable (because their impulse responses would be non-causal and infinitely long), they serve as crucial benchmarks and design specifications for practical filters.

Common Characteristics of Ideal Filters:
1. Passband: A frequency range where the filter's magnitude response is exactly 1 (meaning signals in this band pass through without amplitude change).
2. Stopband: A frequency range where the filter's magnitude response is exactly 0 (meaning signals in this band are completely blocked).
3. Sharp Cutoff: An instantaneous transition between the passband and stopband.
4. Linear Phase: Typically assumed to have a linear phase response in the passband to ensure no phase distortion.

Types of Ideal Filters:
1. Ideal Low-pass Filter (LPF):
- Purpose: To pass low-frequency components and block high-frequency components.
- Magnitude Response: |H(jω)| = 1 for |ω| <= ω_c (cutoff frequency), and 0 for |ω| > ω_c.
- Phase Response: Usually angle(H(jω)) = -kω for |ω| <= ω_c (linear phase in passband), and undefined (or 0) elsewhere due to zero magnitude.
- Effect in Time Domain: Smooths signals by removing high-frequency details. Its impulse response in the time domain is a sinc function.

1. Ideal High-pass Filter (HPF):
2. Purpose: To block low-frequency components (including DC) and pass high-frequency components.
3. Magnitude Response: |H(jω)| = 0 for |ω| < ω_c, and 1 for |ω| >= ω_c.
4. Phase Response: Linear phase in passband.
5. Effect in Time Domain: Highlights sharp changes and edges in a signal by removing the slowly varying (low-frequency) parts.
6. Ideal Band-pass Filter (BPF):
7. Purpose: To pass frequencies within a specific band and block frequencies outside that band.
8. Magnitude Response: |H(jω)| = 1 for ω_c1 <= |ω| <= ω_c2, and 0 otherwise.
9. Phase Response: Linear phase in passband.
10. Effect in Time Domain: Used to isolate a particular range of frequencies, such as tuning into a specific radio station.
11. Ideal Band-stop Filter (BSF) / Ideal Notch Filter:
12. Purpose: To block frequencies within a specific band and pass frequencies outside that band.
13. Magnitude Response: |H(jω)| = 0 for ω_c1 <= |ω| <= ω_c2, and 1 otherwise.
14. Phase Response: Linear phase in passband.
15. Effect in Time Domain: Used to eliminate unwanted noise that is concentrated at a specific frequency or within a narrow band (e.g., removing a 50 Hz or 60 Hz power line hum).