Since the phase is fixed and the amplitude is normalized, the only possible real part is \( a = \frac{1}{2} \). There are no other values satisfying both the magnitude and phase condition.

["Unlocking Signal Phases: Why ( a = \frac{1}{2} ) Is the Only Real Part When Phase and Amplitude Are Fixed", "In complex signal analysis and Fourier theory, phase and amplitude are fundamental descriptors of sinusoidal components. A common scenario arises in linear systems and harmonic decomposition: when the phase is fixed and the amplitude is normalized, only one valid real-valued component remains — specifically, ( a = \frac{1}{2} ). But why is this the case? Let’s explore the mathematical constraints that limit this real part to a single solution.", "### Understanding Phase and Amplitude in Complex Signals", "Any complex-valued sinusoidal function can be expressed as:", "[\nx(t) = a e^{i\omega t + i\phi} = a \cos(\omega t + \phi)\n]", "where:\n- ( a ) is the normalized amplitude (non-negative real),\n- ( \phi ) is the fixed phase angle,\n- The real part is ( \Re[x(t)] = a \cos(\omega t + \phi) ).", "When the amplitude is normalized to 1 (or scaled so ( a = 1 )) and the phase ( \phi ) is fixed, the signal becomes a normalized complex sinusoid. However, in many physical and engineering models, the amplitude is fixed to ( a = \frac{1}{2} ), not 1, introducing a precise constraint.", "### The Constraint: Fixed Phase and Normalized Amplitude", "Suppose:\n- The amplitude is fixed at ( a = \frac{1}{2} ), so the magnitude of the complex representation is ( \left| a \right| = \frac{1}{2} ),\n- The phase ( \phi ) is specified and immutable.", "Then the real part is:", "[\n\Re[x(t)] = \frac{1}{2} \cos(\omega t + \phi)\n]", "The only requirement here is consistency — how many real numbers ( a ) satisfy ( |a| = \frac{1}{2} ) under the phase constraint?", "### Why Only One Real Part Fits the Conditions?", "Mathematically, normalization conventionally fixes the amplitude magnitude to 1, but here ( a ) is explicitly normalized such that the real sinusoid’s envelope matches amplitude ( \frac{1}{2} ). This means the normalized amplitude parameter ( a ) must equal ( \frac{1}{2} ) exactly — no deviation.", "However, the phase ( \phi ) is not a variable to be varied; it’s fixed by physical context or system design. Thus, with:\n- ( |a| = \frac{1}{2} ),\n- ( \phi = \ ext{constant} ),", "there is only one consistent real-valued component:", "[\n\Re[x(t)] = \frac{1}{2} \cos(\omega t + \phi)\n]", "Any other real number ( a ) with ( |a| = \frac{1}{2} ) (i.e., ( a = \pm \frac{1}{2} )) would either violate phase consistency or amplitude scaling unless the phase adapts — but this contradicts the premise of fixed phase and fixed amplitude.", "In other words:\n- Fixing amplitude magnitude uniquely pins ( a = \frac{1}{2} ) (since we’re working in real numbers, and phase shift doesn’t alter amplitude),\n- Fixing phase removes freedom to scale differently,\n- Thus, only one real coefficient ( a = \frac{1}{2} ) produces a valid solution meeting both constraints.", "### Practical Implications in Signal Processing", "This mathematical rigidity ensures stability and consistency in applications such as:\n- Harmonic balancing in power systems,\n- Signal filtering algorithms with phase constraints,\n- Digital signal processing where phase-stabilized modulation is critical.", "Deviating from ( a = \frac{1}{2} ) under fixed phase breaks amplitude scaling and risks signal misrepresentation.", "### Summary: The Uniqueness of ( a = \frac{1}{2} )", "When the amplitude is normalized to ( \frac{1}{2} ) and the phase is fixed, the system admits only one valid real-valued component:", "[\n\boxed{a = \frac{1}{2}}\n]", "This uniqueness stems from the double constraint — fixed amplitude magnitude and fixed phase — leaving no flexibility in the real coefficient. Recognizing this principle ensures accurate modeling, analysis, and synthesis of harmonic and periodic signals in both theory and application.", "---", "Keywords:\ncomplex signal, phase fixed, amplitude normalized, real part only, ( a = \frac{1}{2} ), Fourier analysis, harmonic decomposition, signal processing, signal constraints, phase-examped sinusoid."]