{"id":94066,"date":"2026-08-13T16:41:09","date_gmt":"2026-08-13T08:41:09","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/94066.html"},"modified":"2026-08-13T16:41:09","modified_gmt":"2026-08-13T08:41:09","slug":"%e3%80%90%e9%ab%98%e7%ba%a7%e6%95%b0%e5%ad%97%e4%bf%a1%e5%8f%b7%e5%a4%84%e7%90%86%e3%80%91%e8%b6%85%e8%af%a6%e7%bb%86%e6%8c%87%e5%8d%97lab1b%ef%bc%9amatlab-%e9%a2%91%e5%9f%9f%e6%93%8d%e6%8e%a7","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/94066.html","title":{"rendered":"\u3010\u9ad8\u7ea7\u6570\u5b57\u4fe1\u53f7\u5904\u7406\u3011\u8d85\u8be6\u7ec6\u6307\u5357lab1b\uff1aMATLAB \u9891\u57df\u64cd\u63a7\u4e0e\u4fe1\u53f7\u6062\u590d \u2014\u2014 \u4ece\u96f6\u586b\u5145\u4fe1\u53f7\u5230\u524a\u6ce2\u57fa\u6ce2\u63d0\u53d6\u3010\u542bmatlab\u4ee3\u7801\u3011"},"content":{"rendered":"

\u8d85\u8be6\u7ec6\u6307\u5357：MATLAB \u9891\u57df\u64cd\u63a7\u4e0e\u4fe1\u53f7\u6062\u590d \u2014\u2014 \u4ece\u96f6\u586b\u5145\u4fe1\u53f7\u5230\u524a\u6ce2\u57fa\u6ce2\u63d0\u53d6<\/h2>\n

Ultra-Detailed Guide: Frequency-Domain Manipulation and Signal Recovery in MATLAB \u2014 From Zero-Padded Signals to Fundamental Extraction from Clipped Waveforms<\/p>\n

\n

🌐 English Version (Ultra-Detailed)<\/h3>\n

📌 Article Abstract (Directly Searchable)<\/h4>\n

This advanced guide focuses on inverse DFT operations \u2014 manipulating spectra to recover or transform time-domain signals. It addresses three classic tasks:<\/p>\n

  • The reshape analogy: Reshaping a 10\u00d7128 matrix (with a cosine in row 1) into a 1\u00d71280 vector to mimic a \u201cburst\u201d signal, comparing it to analog low-duty-cycle sampling, and analyzing its interpolated FFT spectrum.<\/li>\n
  • Extracting a continuous sinusoid from a zero-padded signal: Selecting only the conjugate frequency bins of the original tone, zeroing all others, and performing IFFT to obtain a continuous sine wave without inserted zeros \u2014 with quantitative amplitude derivation.<\/li>\n
  • Recovering the fundamental from a clipped signal: Using the identical bin-selection technique to extract the 2.25kHz fundamental from the previously generated clipped waveform, while also dissecting common coding mistakes (e.g., zeroing a wide band, using clipping ratio incorrectly) and providing the correct implementation.<\/li>\n
    \n

    \"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"
    \n\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\"<\/p>\n

    🛠\ufe0f Zero: Mathematical Prerequisites<\/h4>\n
      \n
    • Conjugate Symmetry: For real signals, ( X[k] = X^*[N-k] ). Both positive and negative frequency bins must be preserved together.<\/li>\n
    • DFT Amplitude Scaling: For a pure sine of amplitude ( A ) and length ( N ), the positive-frequency FFT magnitude is ( A \\\\cdot N \/ 2 ). Zero-padding to length ( N_2 ) scales this value to ( A \\\\cdot N_2 \/ 2 ).<\/li>\n<\/ul>\n
      \n

      📝 Part 1: The reshape Magic & Sampling Analogy (Q1)<\/h4>\n

      Task: Run the given code, observe the waveform, and explain its FFT.<\/p>\n

      Code dissection:<\/p>\n

        \n
      • z is a 2Hz cosine sampled at 128Hz (128 points).<\/li>\n
      • x = zeros(10,128); x(1,:)=z; creates a matrix with only the first row filled.<\/li>\n
      • reshape(x,1,1280) reads column by column. It outputs: [z(1), 0,0…0 (9 zeros), z(2), 0,0…0, …].<\/li>\n
      • Result: A 1-second burst of 2Hz cosine followed by 9 seconds of silence.<\/li>\n<\/ul>\n

        Analogy: This resembles pulse sampling or gated sampling in analog systems \u2014 the signal is visible only during a short duty cycle.<\/p>\n

        FFT Analysis: The spectrum shows a sinc-shaped main lobe centered at 2Hz, with interpolated side lobes due to zero-padding (10\u00d7 higher frequency resolution).<\/p>\n

        \n

        📊 Part 2: Extracting a Continuous Sine (No Zero Insertion) \u2014 Q2<\/h4>\n

        Task: Manipulate the FFT to recover a sine wave that is continuous over the whole time axis (no zeros), and explain its amplitude.<\/p>\n

        Correct Approach:<\/p>\n

          \n
        • Locate the exact positive and negative frequency bins for 2Hz in the 1280-point FFT.<\/li>\n
        • Zero-frequency indexing: 2Hz corresponds to normalized index ( 2\/128 ). In 1280-point FFT, ( k = (2\/128) \\\\times 1280 = 20 ) (0\u2011based), so MATLAB index = 21. Negative conjugate index = ( 1280 – 20 + 2 = 1262 ).<\/li>\n<\/ul>\n

          Correct Code:<\/p>\n

          X =<\/span> fft<\/span>(<\/span>x)<\/span>;<\/span>
          \nX_clean =<\/span> zeros<\/span>(<\/span>1<\/span>,<\/span>1280<\/span>)<\/span>;<\/span>
          \nX_clean<\/span>(<\/span>21<\/span>)<\/span> =<\/span> X<\/span>(<\/span>21<\/span>)<\/span>;<\/span>
          \nX_clean<\/span>(<\/span>1262<\/span>)<\/span> =<\/span> X<\/span>(<\/span>1262<\/span>)<\/span>;<\/span>
          \ny =<\/span> real<\/span>(<\/span>ifft<\/span>(<\/span>X_clean)<\/span>)<\/span>;<\/span><\/p>\n

          Amplitude Derivation (Crucial):<\/p>\n

            \n
          • Original 128-point FFT magnitude at 2Hz = ( 1 \\\\times 128 \/ 2 = 64 ).<\/li>\n
          • After zero-padding to 1280, the magnitude scales to ( 64 \\\\times (1280\/128) = 640 ).<\/li>\n
          • IFFT with one conjugate pair yields amplitude = ( 2 \\\\times 640 \/ 1280 = 1.0 ).<\/li>\n
          • Conclusion: The recovered continuous sine has an amplitude of exactly 1.0, matching the original.<\/li>\n<\/ul>\n
            \n

            🎯 Part 3: Fundamental Extraction from Clipped Signal \u2014 Q3 (Real Recovery)<\/h4>\n

            Task: From the 2.25kHz clipped signal (fs=5kHz, clipped to \u00b10.3), extract a clean sinusoid at the original frequency.<\/p>\n

            Feasibility: The clipped spectrum contains the fundamental (2.25kHz) and aliased harmonics (1.75k, 1.25k, etc.). They occupy distinct bins (resolution = 1Hz). We keep only the fundamental bin and discard all others.<\/p>\n

            Correct Code:<\/p>\n

            X =<\/span> fft<\/span>(<\/span>signal)<\/span>;<\/span> N =<\/span> length<\/span>(<\/span>X)<\/span>;<\/span> % N=5000<\/span>
            \nk_fund =<\/span> round<\/span>(<\/span>2250<\/span> *<\/span> 5000<\/span> \/<\/span> 5000<\/span>)<\/span> +<\/span> 1<\/span>;<\/span> % = 2251<\/span>
            \nk_neg =<\/span> N –<\/span> k_fund +<\/span> 2<\/span>;<\/span> % = 2751<\/span>
            \nX_clean =<\/span> zeros<\/span>(<\/span>1<\/span>,<\/span>N)<\/span>;<\/span>
            \nX_clean<\/span>(<\/span>k_fund)<\/span> =<\/span> X<\/span>(<\/span>k_fund)<\/span>;<\/span>
            \nX_clean<\/span>(<\/span>k_neg)<\/span> =<\/span> X<\/span>(<\/span>k_neg)<\/span>;<\/span>
            \nrecovered =<\/span> real<\/span>(<\/span>ifft<\/span>(<\/span>X_clean)<\/span>)<\/span>;<\/span><\/p>\n

            Result Analysis:<\/p>\n

              \n
            • Waveform: Perfectly smooth sine at 2.25kHz.<\/li>\n
            • Amplitude: Approximately 0.7 (not 1.0). Why? Because clipping transferred part of the fundamental energy to harmonics. We extracted only the remaining fundamental component. To restore the original 1.0 amplitude, one would need prior knowledge of the clipping level and use nonlinear compensation (e.g., iterative algorithms), which is beyond linear frequency-domain bin selection.<\/li>\n<\/ul>\n
              \n

              🔍 Part 4: Dissecting Common Coding Mistakes (from your .m files)<\/h4>\n

              Mistake 1: Zeroing a Wide Frequency Band (lab1bb.m Plan2)<\/p>\n

              clipping_range_indices =<\/span> round<\/span>(<\/span>[<\/span>1000<\/span>,<\/span>3000<\/span>]<\/span>\/<\/span>freq_res)<\/span>+<\/span>1<\/span>;<\/span>
              \nX<\/span>(<\/span>clipping_range_indices<\/span>(<\/span>1<\/span>)<\/span>:<\/span>clipping_range_indices<\/span>(<\/span>2<\/span>)<\/span>)<\/span> =<\/span> …<\/span>
              \n abs<\/span>(<\/span>X<\/span>(<\/span>clipping_range_indices<\/span>(<\/span>1<\/span>)<\/span>:<\/span>clipping_range_indices<\/span>(<\/span>2<\/span>)<\/span>)<\/span>)<\/span>;<\/span><\/p>\n

              Why wrong: This attempts a band-stop\/band-modify filter, which corrupts phase and affects the fundamental itself (2250Hz lies inside 1000~3000Hz).<\/p>\n

              Mistake 2: Using Clipping Ratio to Define Cutoff (lab1b.m)<\/p>\n

              cut_ratio =<\/span> 0.3<\/span> \/<\/span> 1<\/span>;<\/span>
              \nN_cut =<\/span> round<\/span>(<\/span>N\/<\/span>2<\/span>)<\/span> *<\/span> cut_ratio;<\/span>
              \nX_cut<\/span>(<\/span>N_cut+<\/span>1<\/span>:<\/span>end<\/span>)<\/span> =<\/span> 0<\/span>;<\/span><\/p>\n

              Why wrong: The clipping ratio determines time-domain peak truncation, not the frequency positions of harmonics. Aliased frequencies (1.75k, 1.25k) are determined by ( f_s ) and ( f ), not by the clipping threshold. This operation is essentially low-pass filtering, which cannot isolate the fundamental from in-band aliases.<\/p>\n

              Mistake 3: Operating on abs after fftshift (lab1bb2.m)<\/p>\n

              y2 =<\/span> fftshift<\/span>(<\/span>abs<\/span>(<\/span>fft<\/span>(<\/span>signal)<\/span>)<\/span>)<\/span>;<\/span>
              \ny2<\/span>(<\/span>(<\/span>5001<\/span>–<\/span>1<\/span>)<\/span>*<\/span>0.1<\/span>:<\/span>(<\/span>5001<\/span>–<\/span>1<\/span>)<\/span>*<\/span>0.9<\/span>)<\/span> =<\/span> 0<\/span>;<\/span><\/p>\n

              Why wrong: abs destroys phase information. Zeroing a central band acts as a strange notch filter and cannot extract a single tone.<\/p>\n

              The Golden Rule: To extract a specific frequency, operate on a single complex bin (and its conjugate), not on a range of bins. This is equivalent to an ideal narrow-band filter with bandwidth ( \\\\Delta f ).<\/p>\n

              \n

              💎 Summary and Engineering Insights<\/h4>\n\nScenarioCorrect MethodRecovered AmplitudeKey Constraint<\/tr>\n\n\n\n\n
              Recover continuous sine from zero-padded burst<\/td>\nKeep only the conjugate bins of the tone<\/td>\nEqual to original (1.0)<\/td>\nIFFT length must match FFT length<\/td>\n<\/tr>\n
              Recover fundamental from clipped signal<\/td>\nKeep only the fundamental conjugate bins<\/td>\nEqual to post-clipping fundamental amplitude (<1.0)<\/td>\nCannot restore lost harmonic energy<\/td>\n<\/tr>\n
              General aliased component separation<\/td>\nPossible only if target and alias are in different bins<\/td>\nDepends on the actual component energy<\/td>\nIf they share the same bin, separation is mathematically impossible<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

              Engineering Takeaway: Frequency-domain bin selection is a powerful tool for extracting known periodic components from distorted signals, widely used in power quality monitoring (fundamental extraction), ECG signal processing, and narrowband interference suppression. However, it fails when the sampling rate is too low to resolve the target and interfering frequencies into distinct bins.<\/p>\n

              📌 \u6587\u7ae0\u7b80\u4ecb（\u53ef\u76f4\u63a5\u68c0\u7d22\u5230\u9898\u76ee）<\/h4>\n

              \u672c\u6587\u662f\u201c\u4fe1\u53f7\u91c7\u6837\u4e0e\u9891\u8c31\u5206\u6790\u201d\u7cfb\u5217\u7684\u8fdb\u9636\u7bc7，\u805a\u7126\u4e8e \u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362（DFT）\u7684\u9006\u64cd\u4f5c，\u5373\u5982\u4f55\u901a\u8fc7\u64cd\u63a7\u9891\u8c31\u6765\u201c\u6539\u9020\u201d\u6216\u201c\u6062\u590d\u201d\u65f6\u57df\u4fe1\u53f7\u3002\u5185\u5bb9\u4e25\u683c\u5bf9\u5e94\u4ee5\u4e0b\u4e09\u9053\u7ecf\u5178\u4efb\u52a1：<\/p>\n

            • reshape \u4e0e\u201c\u91c7\u6837\u201d\u7c7b\u6bd4：\u5c06\u4e00\u4e2a 10\u00d7128 \u7684\u77e9\u9635（\u4ec5\u7b2c\u4e00\u884c\u5b58\u653e 2Hz \u4f59\u5f26\u6ce2）\u91cd\u5851\u4e3a 1\u00d71280 \u7684\u957f\u5411\u91cf，\u89c2\u5bdf\u5176\u201c\u7a81\u53d1\u201d\u6ce2\u5f62，\u5e76\u4e0e\u6a21\u62df\u57df\u4e2d\u7684\u4f4e\u5360\u7a7a\u6bd4\u91c7\u6837\u4fe1\u53f7\u8fdb\u884c\u7c7b\u6bd4，\u540c\u65f6\u5206\u6790\u5176 FFT \u4e3a\u4f55\u5448\u73b0\u63d2\u503c\u72b6\u9891\u8c31\u3002<\/li>\n
            • \u4ece\u8865\u96f6\u4fe1\u53f7\u4e2d\u63d0\u53d6\u8fde\u7eed\u6b63\u5f26\u6ce2：\u901a\u8fc7\u5bf9 FFT \u8fdb\u884c\u201c\u9891\u70b9\u7b5b\u9009\u201d（\u53ea\u4fdd\u7559\u539f\u9891\u7387\u5bf9\u5e94\u7684\u6b63\u8d1f\u5171\u8f6d\u9891\u70b9，\u5176\u4f59\u7f6e\u96f6），\u518d\u6267\u884c IFFT，\u5f97\u5230\u4e00\u4e2a\u5728\u6574\u4e2a\u65f6\u95f4\u8f74\u4e0a\u8fde\u7eed\u632f\u8361（\u65e0\u63d2\u5165\u96f6）\u7684\u6b63\u5f26\u6ce2，\u5e76\u5b9a\u91cf\u63a8\u5bfc\u5176\u5e45\u5ea6\u4e3a\u4ec0\u4e48\u7b49\u4e8e\u539f\u59cb\u5e45\u5ea6 1.0\u3002<\/li>\n
            • \u4ece\u524a\u6ce2\u4fe1\u53f7\u4e2d\u6062\u590d\u539f\u59cb\u9891\u7387\u7684\u7eaf\u51c0\u6b63\u5f26\u6ce2：\u5229\u7528\u5b8c\u5168\u76f8\u540c\u7684\u9891\u57df\u7b5b\u9009\u6280\u672f，\u4ece\u4e4b\u524d\u751f\u6210\u7684 2.25kHz \u524a\u6ce2\u4fe1\u53f7（\u91c7\u6837\u7387 5kHz）\u4e2d\u63d0\u53d6\u51fa\u57fa\u6ce2，\u5b9e\u73b0\u201c\u524a\u6ce2\u6062\u590d\u201d\u3002\u540c\u65f6\u91cd\u70b9\u5256\u6790\u5b66\u751f\u4ee3\u7801\u4e2d\u7684\u5178\u578b\u9519\u8bef（\u5982\u9519\u8bef\u5730\u7f6e\u96f6\u4e00\u6bb5\u9891\u5e26\u3001\u9519\u8bef\u5730\u4f7f\u7528\u524a\u6ce2\u6bd4\u4f8b\u7b49），\u5e76\u7ed9\u51fa\u6b63\u786e\u7684\u5b9e\u73b0\u65b9\u6cd5\u3002<\/li>\n
              \n

              🔬 \u96f6\u3001\u9884\u5907\u77e5\u8bc6\u4e0e\u6838\u5fc3\u6570\u5b66\u539f\u7406<\/h4>\n

              \u5728\u5f00\u59cb\u64cd\u4f5c\u4e4b\u524d，\u5fc5\u987b\u6df1\u523b\u7406\u89e3\u4ee5\u4e0b\u4e24\u6761 DFT \u6027\u8d28：<\/p>\n

                \n
              • \u5171\u8f6d\u5bf9\u79f0\u6027：\u5bf9\u4e8e\u5b9e\u4fe1\u53f7，\u5176 DFT \u9891\u8c31\u6ee1\u8db3 ( X[k] = X^*[N – k] )（\u5176\u4e2d ( N ) \u4e3a DFT \u957f\u5ea6）\u3002\u56e0\u6b64，\u4fdd\u7559\u4e00\u4e2a\u6b63\u9891\u7387\u5206\u91cf，\u5fc5\u987b\u540c\u65f6\u4fdd\u7559\u5176\u5bf9\u5e94\u7684\u8d1f\u9891\u7387\u5171\u8f6d\u5206\u91cf，\u5426\u5219 IFFT \u540e\u4f1a\u4ea7\u751f\u590d\u6570\u4fe1\u53f7（\u865a\u90e8\u4e0d\u4e3a\u96f6）\u3002<\/li>\n
              • DFT \u5e45\u5ea6\u4e0e\u6b63\u5f26\u6ce2\u5e45\u5ea6\u7684\u5173\u7cfb：\u5bf9\u4e8e\u4e00\u4e2a\u957f\u5ea6\u4e3a ( N )\u3001\u5e45\u5ea6\u4e3a ( A ) \u7684\u7eaf\u6b63\u5f26\u6ce2 ( A \\\\cos(2\\\\pi f t) )，\u5176\u5355\u8fb9\u672a\u5f52\u4e00\u5316 FFT \u5728\u6b63\u9891\u7387\u5904\u7684\u5e45\u5ea6\u4e3a ( A \\\\cdot N \/ 2 )\u3002\u82e5\u5c06\u9891\u8c31\u957f\u5ea6\u4ece ( N_1 ) \u8865\u96f6\u5ef6\u957f\u5230 ( N_2 )，\u5219\u8be5\u9891\u70b9\u7684\u5e45\u5ea6\u503c\u4f1a\u6309\u6bd4\u4f8b\u53d8\u4e3a ( A \\\\cdot N_2 \/ 2 )（\u56e0\u4e3a FFT \u7b97\u6cd5\u9ed8\u8ba4\u4e0d\u505a\u80fd\u91cf\u5f52\u4e00\u5316）\u3002<\/li>\n<\/ul>\n
                \n

                📝 \u7b2c\u4e00\u90e8\u5206：reshape \u7684\u9b54\u672f\u4e0e\u201c\u91c7\u6837\u201d\u7c7b\u6bd4（\u5bf9\u5e94 Q1）<\/h4>\n

                \u4efb\u52a1\u63cf\u8ff0<\/h5>\n

                \u8fd0\u884c\u4ee5\u4e0b MATLAB \u4ee3\u7801，\u89c2\u5bdf Figure 1 \u4e2d\u7684\u4fe1\u53f7\u201c\u770b\u8d77\u6765\u50cf\u4ec0\u4e48\u201d\u2014\u2014\u6362\u53e5\u8bdd\u8bf4，\u5b83\u5728\u54ea\u4e9b\u65b9\u9762\u7c7b\u4f3c\u4e8e\u6a21\u62df\u57df\u4e2d\u7684\u91c7\u6837\u4fe1\u53f7？\u653e\u5927\u89c2\u5bdf，\u5e76\u5bf9 x \u8fdb\u884c FFT，\u89e3\u91ca\u4f60\u770b\u5230\u7684\u73b0\u8c61\u3002<\/p>\n

                x =<\/span> zeros<\/span>(<\/span>10<\/span>,<\/span> 128<\/span>)<\/span>;<\/span>
                \nt1 =<\/span> 0<\/span>:<\/span>1<\/span>\/<\/span>128<\/span>:<\/span>1<\/span>–<\/span>1<\/span>\/<\/span>128<\/span>;<\/span>
                \nz =<\/span> cos<\/span>(<\/span>2<\/span> *<\/span> pi<\/span> *<\/span> 2<\/span> *<\/span> t1)<\/span>;<\/span> % 2Hz \u4f59\u5f26，128 \u4e2a\u91c7\u6837\u70b9（\u91c7\u6837\u7387 128Hz）<\/span>
                \nx<\/span>(<\/span>1<\/span>,<\/span> :<\/span>)<\/span> =<\/span> z;<\/span> % \u7b2c\u4e00\u884c\u5b58\u5165\u4f59\u5f26\u6ce2<\/span>
                \nx =<\/span> reshape<\/span>(<\/span>x,<\/span> 1<\/span>,<\/span> 1280<\/span>)<\/span>;<\/span> % \u5c06 10\u00d7128 \u77e9\u9635\u91cd\u5851\u4e3a 1\u00d71280 \u7684\u884c\u5411\u91cf<\/span>
                \nfigure<\/span>(<\/span>1<\/span>)<\/span>;<\/span> plot<\/span>(<\/span>x)<\/span>;<\/span><\/p>\n

                \u4ee3\u7801\u9010\u6b65\u62c6\u89e3\u4e0e\u7269\u7406\u610f\u4e49<\/h5>\n
              • \n

                \u751f\u6210\u57fa\u7840\u4fe1\u53f7 z：<\/p>\n

                  \n
                • \u91c7\u6837\u7387 ( f_{s1} = 128 \\\\text{ Hz} )，\u65f6\u957f 1 \u79d2，\u5171 128 \u4e2a\u70b9\u3002<\/li>\n
                • \u4fe1\u53f7\u9891\u7387\u4e3a 2 Hz，\u6bcf\u4e2a\u5468\u671f\u5305\u542b ( 128 \/ 2 = 64 ) \u4e2a\u91c7\u6837\u70b9，\u6ce2\u5f62\u975e\u5e38\u5149\u6ed1\u3002<\/li>\n<\/ul>\n<\/li>\n
                • \n

                  \u6784\u9020\u77e9\u9635 x：<\/p>\n

                    \n
                  • zeros(10, 128) \u751f\u6210\u4e86\u4e00\u4e2a 10 \u884c\u3001128 \u5217\u7684\u5168\u96f6\u77e9\u9635\u3002<\/li>\n
                  • \u5c06 z \u8d4b\u503c\u7ed9\u7b2c\u4e00\u884c（\u7b2c 2~10 \u884c\u4fdd\u6301\u5168\u96f6）\u3002<\/li>\n<\/ul>\n<\/li>\n
                  • \n

                    reshape \u64cd\u4f5c（\u5173\u952e）：<\/p>\n

                      \n
                    • MATLAB \u7684 reshape \u6309\u5217\u4f18\u5148\u7684\u987a\u5e8f\u91cd\u6392\u5143\u7d20\u3002<\/li>\n
                    • \u539f\u59cb 10\u00d7128 \u77e9\u9635\u6709 1280 \u4e2a\u5143\u7d20\u3002reshape \u5c06\u5b83\u4eec\u9010\u4e00\u53d6\u51fa，\u4f9d\u6b21\u586b\u5165 1\u00d71280 \u7684\u884c\u5411\u91cf\u4e2d\u3002<\/li>\n
                    • \u987a\u5e8f\u89e3\u6790：\n
                        \n
                      • \u5148\u53d6\u7b2c 1 \u5217：\u7b2c 1 \u884c\u662f z(1)，\u7b2c 2~10 \u884c\u662f 0 \u2192 \u8f93\u51fa [z(1), 0, 0, …, 0]（\u5171 10 \u4e2a\u5143\u7d20）\u3002<\/li>\n
                      • \u518d\u53d6\u7b2c 2 \u5217：[z(2), 0, 0, …, 0]\u3002<\/li>\n
                      • \u4ee5\u6b64\u7c7b\u63a8，\u76f4\u5230\u7b2c 128 \u5217\u3002<\/li>\n<\/ul>\n<\/li>\n
                      • \u6700\u7ec8\u7ed3\u679c：\u8f93\u51fa\u7684 1\u00d71280 \u5411\u91cf\u4e2d，\u524d 128 \u4e2a\u70b9\u662f z（2Hz \u4f59\u5f26），\u7d27\u63a5\u7740\u7684 1152 \u4e2a\u70b9（128\u00d79）\u5168\u90e8\u662f 0！<\/li>\n<\/ul>\n<\/li>\n
                        \u201c\u770b\u8d77\u6765\u50cf\u4ec0\u4e48\u201d？\u2014\u2014\u4e0e\u6a21\u62df\u57df\u91c7\u6837\u7684\u7c7b\u6bd4<\/h5>\n
                          \n
                        • \u65f6\u57df\u56fe\u50cf\u663e\u793a：\u4e00\u4e2a 2Hz \u7684\u4f59\u5f26\u6ce2\u5305（\u6301\u7eed 1 \u79d2），\u540e\u9762\u7d27\u8ddf\u7740\u4e00\u6bb5 9 \u79d2\u957f\u7684\u9759\u9ed8（\u96f6\u503c）\u3002<\/li>\n
                        • \u8fd9\u4e0e\u6a21\u62df\u57df\u4e2d\u7684 \u201c\u8109\u51b2\u91c7\u6837\u201d\u6216\u201c\u9009\u901a\u91c7\u6837\u201d \u975e\u5e38\u7c7b\u4f3c：\u4fe1\u53f7\u53ea\u5728\u5f88\u77ed\u7684\u201c\u65f6\u95f4\u7a97\u53e3\u201d\u5185\u51fa\u73b0（\u5360\u7a7a\u6bd4 1\/10），\u5176\u4f59\u65f6\u95f4\u88ab\u5f3a\u5236\u5f52\u96f6\u3002\u8fd9\u79cd\u64cd\u4f5c\u5728\u96f7\u8fbe\u3001\u8d85\u58f0\u6210\u50cf\u7b49\u7cfb\u7edf\u4e2d\u5e38\u7528\u6765\u6a21\u62df\u201c\u7a81\u53d1\u4fe1\u53f7\u201d（Burst Signal）\u3002<\/li>\n<\/ul>\n
                          FFT \u5206\u6790\u4e0e\u89e3\u91ca<\/h5>\n

                          figure<\/span>(<\/span>1<\/span>)<\/span>;<\/span>
                          \nsubplot<\/span>(<\/span>2<\/span>,<\/span>1<\/span>,<\/span>2<\/span>)<\/span>;<\/span>
                          \nplot<\/span>(<\/span>fftshift<\/span>(<\/span>abs<\/span>(<\/span>fft<\/span>(<\/span>x)<\/span>)<\/span>)<\/span>)<\/span>;<\/span><\/p>\n

                          \u9891\u8c31\u7279\u5f81：<\/p>\n

                        • \u4e3b\u74e3：\u5728\u5bf9\u5e94\u4e8e 2 Hz \u7684\u9891\u7387\u4f4d\u7f6e\u51fa\u73b0\u5cf0\u503c（\u7531\u4e8e\u8865\u96f6\u6269\u5c55\u5230 1280 \u70b9，\u9891\u7387\u5206\u8fa8\u7387\u63d0\u9ad8\u4e86 10 \u500d，\u53d8\u4e3a ( 128 \/ 1280 = 0.1 \\\\text{ Hz} )）\u3002<\/li>\n
                        • sinc \u51fd\u6570\u5f62\u72b6：\u56e0\u4e3a\u65f6\u57df\u4fe1\u53f7\u88ab\u77e9\u5f62\u7a97（1 \u79d2\u7a97\u53e3）\u622a\u65ad，\u9891\u57df\u8868\u73b0\u4e3a sinc \u51fd\u6570\u7684\u5f62\u72b6，\u4e3b\u74e3\u4e24\u4fa7\u5b58\u5728\u9010\u6e10\u8870\u51cf\u7684\u65c1\u74e3（\u6805\u74e3）\u3002<\/li>\n
                        • \u63d2\u503c\u6548\u679c：\u539f\u672c 128 \u70b9\u7684 FFT \u53ea\u6709 128 \u4e2a\u9891\u70b9，\u8865\u96f6\u5230 1280 \u70b9\u540e，\u9891\u8c31\u88ab\u201c\u63d2\u503c\u201d\u5f97\u66f4\u5e73\u6ed1，\u80fd\u591f\u66f4\u7cbe\u7ec6\u5730\u663e\u793a sinc \u65c1\u74e3\u7684\u8d77\u4f0f\u3002<\/li>\n
                          \n

                          📊 \u7b2c\u4e8c\u90e8\u5206：\u63d0\u53d6\u201c\u8fde\u7eed\u201d\u65e0\u96f6\u6b63\u5f26\u6ce2（\u5bf9\u5e94 Q2）\u2014\u2014 \u6838\u5fc3\u6062\u590d\u6280\u672f<\/h4>\n

                          \u4efb\u52a1\u63cf\u8ff0<\/h5>\n

                          \u64cd\u63a7\u4e0a\u8ff0\u4fe1\u53f7\u7684 FFT，\u5e76\u6267\u884c IFFT，\u4ee5\u521b\u5efa\u4e00\u4e2a\u5728\u201c\u65f6\u95f4\u201d\u57df\u4e2d\u8fde\u7eed（\u5373\u6ca1\u6709\u5185\u63d2\u96f6\u503c） \u7684\u6b63\u5f26\u6ce2\u3002\u89e3\u91ca\u8be5\u6b63\u5f26\u6ce2\u7684\u5e45\u5ea6\u3002<\/p>\n

                          \u6b63\u786e\u601d\u8def（\u533a\u522b\u4e8e\u9519\u8bef\u4ee3\u7801）<\/h5>\n

                          \u8865\u96f6\u4fe1\u53f7 x \u7684\u9891\u8c31\u4e2d，\u9664\u4e86 2 Hz \u5bf9\u5e94\u7684\u90a3\u6839\u8c31\u7ebf\u5916，\u5176\u4f59\u90fd\u662f sinc \u65c1\u74e3\u548c\u96f6\u503c\u3002\u5982\u679c\u6211\u4eec\u53ea\u4fdd\u7559 2 Hz \u90a3\u6839\u201c\u7eaf\u97f3\u201d\u8c31\u7ebf（\u53ca\u5176\u5171\u8f6d），\u4e22\u5f03\u6240\u6709\u65c1\u74e3\u548c\u96f6\u503c，\u90a3\u4e48 IFFT \u5c06\u53ea\u6062\u590d\u51fa\u8fde\u7eed\u7684\u7eaf\u6b63\u5f26\u6ce2，\u800c\u4e0d\u4f1a\u6709\u4efb\u4f55\u96f6\u503c\u95f4\u9699\u3002<\/p>\n

                          \u9891\u70b9\u7d22\u5f15\u5b9a\u4f4d（\u6570\u5b66\u63a8\u5bfc，\u6781\u5176\u91cd\u8981）<\/h5>\n
                            \n
                          • \u539f\u59cb\u77ed\u5e8f\u5217（128 \u70b9）：2 Hz \u5bf9\u5e94\u7684\u5f52\u4e00\u5316\u9891\u7387\u4e3a ( 2 \/ 128 )\u3002<\/li>\n
                          • \u8865\u96f6\u540e\u957f\u5e8f\u5217（1280 \u70b9）：\u9891\u7387\u5206\u8fa8\u7387\u53d8\u4e3a ( 1 \/ 1280 )\u3002\u65b0\u7d22\u5f15 ( k ) \u6ee1\u8db3：
                            \n[
                            \nk = \\\\frac{f}{f_s} \\\\times N = \\\\frac{2}{128} \\\\times 1280 = 20
                            \n]
                            \n（\u6ce8\u610f：MATLAB \u7d22\u5f15\u4ece 1 \u5f00\u59cb，\u82e5\u6309 0 \u57fa\u7d22\u5f15\u4e3a 20，\u5219 MATLAB \u7d22\u5f15\u4e3a 21）\u3002<\/li>\n
                          • \u5171\u8f6d\u5bf9\u79f0\u4f4d\u7f6e：( N – k + 2 = 1280 – 20 + 2 = 1262 )（MATLAB \u7d22\u5f15）\u3002<\/li>\n<\/ul>\n
                            \u6b63\u786e\u4ee3\u7801\u5b9e\u73b0（\u7ea0\u6b63\u5e38\u89c1\u9519\u8bef）<\/h5>\n

                            % \u63d0\u53d6\u7eaf\u51c0\u9891\u70b9<\/span>
                            \nX =<\/span> fft<\/span>(<\/span>x)<\/span>;<\/span> % \u957f\u5ea6 1280<\/span>
                            \nk_pos =<\/span> 21<\/span>;<\/span> % \u6b63\u9891\u7387\u7d22\u5f15（\u5bf9\u5e94 2Hz）<\/span>
                            \nk_neg =<\/span> 1280<\/span> –<\/span> 21<\/span> +<\/span> 2<\/span>;<\/span> % \u8d1f\u9891\u7387\u7d22\u5f15（\u5171\u8f6d\u4f4d\u7f6e）<\/span><\/p>\n

                            X_clean =<\/span> zeros<\/span>(<\/span>1<\/span>,<\/span> 1280<\/span>)<\/span>;<\/span>
                            \nX_clean<\/span>(<\/span>k_pos)<\/span> =<\/span> X<\/span>(<\/span>k_pos)<\/span>;<\/span> % \u4fdd\u7559\u6b63\u9891\u590d\u6570\u5e45\u503c<\/span>
                            \nX_clean<\/span>(<\/span>k_neg)<\/span> =<\/span> X<\/span>(<\/span>k_neg)<\/span>;<\/span> % \u4fdd\u7559\u8d1f\u9891\u590d\u6570\u5e45\u503c（MATLAB \u81ea\u52a8\u5171\u8f6d）<\/span><\/p>\n

                            % IFFT \u6062\u590d\u65f6\u57df\u4fe1\u53f7<\/span>
                            \ny_continuous =<\/span> real<\/span>(<\/span>ifft<\/span>(<\/span>X_clean)<\/span>)<\/span>;<\/span><\/p>\n

                            figure<\/span>(<\/span>2<\/span>)<\/span>;<\/span>
                            \nplot<\/span>(<\/span>y_continuous)<\/span>;<\/span>
                            \nxlabel<\/span>(<\/span>'Time (samples)'<\/span>)<\/span>;<\/span> ylabel<\/span>(<\/span>'Amplitude'<\/span>)<\/span>;<\/span>
                            \ntitle<\/span>(<\/span>'\u63d0\u53d6\u51fa\u7684\u8fde\u7eed\u6b63\u5f26\u6ce2（\u65e0\u63d2\u96f6\u503c）'<\/span>)<\/span>;<\/span><\/p>\n

                            \u5e45\u5ea6\u5b9a\u91cf\u63a8\u5bfc（\u8003\u8bd5\/\u7406\u8bba\u91cd\u70b9）<\/h5>\n
                              \n
                            • \u539f\u59cb\u4fe1\u53f7 z = cos(2\u03c0\u00b72\u00b7t)，\u5e45\u5ea6 ( A = 1 )\u3002<\/li>\n
                            • \u5728 128 \u70b9\u7684\u672a\u5f52\u4e00\u5316 FFT \u4e2d，\u6b63\u9891\u70b9\u5e45\u5ea6\u4e3a ( A \\\\times N_1 \/ 2 = 1 \\\\times 128 \/ 2 = 64 )\u3002<\/li>\n
                            • \u8865\u96f6\u540e，FFT \u957f\u5ea6\u53d8\u4e3a ( N_2 = 1280 )\u3002\u7531\u4e8e FFT \u7b97\u6cd5\u662f\u7ebf\u6027\u53d8\u6362，\u8be5\u9891\u70b9\u7684\u5e45\u503c\u4f1a\u6309\u957f\u5ea6\u6bd4\u4f8b\u653e\u5927，\u53d8\u4e3a ( 64 \\\\times (1280 \/ 128) = 64 \\\\times 10 = 640 )\u3002<\/li>\n
                            • \u5728 IFFT \u8fc7\u7a0b\u4e2d，\u4fdd\u7559\u4e00\u5bf9\u5171\u8f6d\u9891\u70b9，\u65f6\u57df\u5e45\u5ea6\u7684\u8ba1\u7b97\u516c\u5f0f\u4e3a：
                              \n[
                              \n\\\\text{Amplitude} = \\\\frac{2 \\\\times |X(k)|}{N_2} = \\\\frac{2 \\\\times 640}{1280} = 1.0
                              \n]<\/li>\n
                            • \u7ed3\u8bba：\u6062\u590d\u5f97\u5230\u7684\u8fde\u7eed\u6b63\u5f26\u6ce2\u5e45\u5ea6\u7cbe\u786e\u7b49\u4e8e 1.0，\u4e0e\u539f\u4fe1\u53f7\u5b8c\u5168\u4e00\u81f4\u3002\u8fd9\u662f\u56e0\u4e3a\u6211\u4eec\u4fdd\u7559\u4e86\u8be5\u9891\u7387\u5206\u91cf\u7684\u5168\u90e8\u80fd\u91cf，\u4e14 IFFT \u7684\u5f52\u4e00\u5316\u7cfb\u6570 ( 1\/N ) \u6070\u597d\u62b5\u6d88\u4e86 FFT \u7684\u957f\u5ea6\u653e\u5927\u6548\u5e94\u3002<\/li>\n<\/ul>\n

                              \u26a0\ufe0f \u91cd\u8981\u533a\u5206：\u6709\u4e9b\u5b66\u751f\u4f1a\u8bd5\u56fe\u901a\u8fc7\u201c\u6ee4\u6ce2\u201d\u4fdd\u7559\u4e00\u4e2a\u9891\u7387\u8303\u56f4（\u5982\u628a\u7d22\u5f15 1~200 \u90fd\u4fdd\u7559），\u4f46\u8fd9\u4f1a\u5f15\u5165 sinc \u65c1\u74e3，\u5bfc\u81f4\u65f6\u57df\u6ce2\u5f62\u51fa\u73b0\u201c\u62d6\u5c3e\u201d\u6216\u5e45\u5ea6\u504f\u5dee\u3002\u5fc5\u987b\u7cbe\u786e\u5230\u5355\u4e2a\u9891\u70b9，\u624d\u80fd\u5b8c\u7f8e\u6062\u590d\u7eaf\u51c0\u5355\u97f3\u3002<\/p>\n

                              \n

                              🎯 \u7b2c\u4e09\u90e8\u5206：\u4ece\u524a\u6ce2\u4fe1\u53f7\u4e2d\u63d0\u53d6\u57fa\u6ce2（\u5bf9\u5e94 Q3）\u2014\u2014 \u771f\u6b63\u7684\u201c\u524a\u6ce2\u6062\u590d\u201d<\/h4>\n

                              \u4efb\u52a1\u63cf\u8ff0<\/h5>\n

                              \u53d6 Lab1A \u7b2c\u4e09\/\u56db\u90e8\u5206\u4e2d\u7684\u524a\u6ce2\u4fe1\u53f7（\u5373 2.25kHz \u6b63\u5f26\u6ce2，\u91c7\u6837\u7387 5kHz，\u786c\u524a\u6ce2\u81f3 \u00b10.3）\u3002\u4f60\u80fd\u4f7f\u7528\u4e0e\u4e0a\u9762\u7c7b\u4f3c\u7684\u6280\u672f（\u9891\u57df\u7b5b\u9009）\u63d0\u53d6\u51fa\u539f\u59cb\u9891\u7387\u5904\u7684\u201c\u5e72\u51c0\u201d\u6b63\u5f26\u6ce2\u5417？<\/p>\n

                              \u53ef\u884c\u6027\u5206\u6790（\u4e3a\u4ec0\u4e48\u8fd9\u6b21\u4e5f\u80fd\u6210\u529f？）<\/h5>\n
                                \n
                              • \u524a\u6ce2\u4fe1\u53f7\u7684\u9891\u8c31\u5305\u542b：\u57fa\u6ce2（2.25kHz）+ \u6df7\u53e0\u8c10\u6ce2（1.75k, 1.25k, 0.75k, 0.25k\u2026）\u3002<\/li>\n
                              • \u5728\u79bb\u6563\u9891\u7387\u8f74\u4e0a，\u8fd9\u4e9b\u6210\u5206\u5360\u636e\u5b8c\u5168\u4e0d\u540c\u7684\u72ec\u7acb\u9891\u70b9（\u56e0\u4e3a\u9891\u7387\u5206\u8fa8\u7387 ( \\\\Delta f = 1 \\\\text{Hz} )，\u5b83\u4eec\u4e4b\u95f4\u76f8\u9694\u6570\u767e Hz）\u3002<\/li>\n
                              • \u56e0\u6b64，\u6211\u4eec\u53ea\u9700\u5b9a\u4f4d\u57fa\u6ce2（2.25kHz）\u5bf9\u5e94\u7684\u90a3\u6839\u8c31\u7ebf，\u628a\u5176\u4ed6\u6240\u6709\u8c31\u7ebf\u201c\u6390\u6389\u201d，\u518d IFFT，\u5c31\u80fd\u5f97\u5230\u53ea\u542b 2.25kHz \u7684\u7eaf\u51c0\u6b63\u5f26\u6ce2\u3002<\/li>\n<\/ul>\n
                                \u6b63\u786e\u4ee3\u7801\u5b9e\u73b0<\/h5>\n

                                % \u590d\u7528 Lab1A \u7684\u524a\u6ce2\u4fe1\u53f7\u751f\u6210\u4ee3\u7801<\/span>
                                \nfrequency =<\/span> 2250<\/span>;<\/span>
                                \nsampling_rate =<\/span> 5000<\/span>;<\/span>
                                \nduration =<\/span> 1<\/span>;<\/span>
                                \nmax_amp =<\/span> 0.3<\/span>;<\/span>
                                \nnum_samples =<\/span> duration *<\/span> sampling_rate;<\/span>
                                \ntime =<\/span> (<\/span>0<\/span>:<\/span>num_samples–<\/span>1<\/span>)<\/span> \/<\/span> sampling_rate;<\/span>
                                \nsignal =<\/span> sin<\/span>(<\/span>2<\/span> *<\/span> pi<\/span> *<\/span> frequency *<\/span> time)<\/span>;<\/span>
                                \nsignal<\/span>(<\/span>signal ><\/span> max_amp)<\/span> =<\/span> max_amp;<\/span>
                                \nsignal<\/span>(<\/span>signal <<\/span> –<\/span>max_amp)<\/span> =<\/span> –<\/span>max_amp;<\/span><\/p>\n

                                % 1. FFT<\/span>
                                \nX =<\/span> fft<\/span>(<\/span>signal)<\/span>;<\/span>
                                \nN =<\/span> length<\/span>(<\/span>X)<\/span>;<\/span> % N = 5000<\/span><\/p>\n

                                % 2. \u8ba1\u7b97\u57fa\u6ce2\u5bf9\u5e94\u7684\u6b63\u9891\u7387\u7d22\u5f15（0Hz \u5bf9\u5e94\u7d22\u5f15 1）<\/span>
                                \n% \u516c\u5f0f：k = round(f * N \/ fs) + 1<\/span>
                                \nk_fund =<\/span> round<\/span>(<\/span>frequency *<\/span> N \/<\/span> sampling_rate)<\/span> +<\/span> 1<\/span>;<\/span>
                                \n% 2250 * 5000 \/ 5000 = 2250，+1 = 2251<\/span><\/p>\n

                                % 3. \u8d1f\u9891\u7387\u5171\u8f6d\u7d22\u5f15<\/span>
                                \nk_neg =<\/span> N –<\/span> k_fund +<\/span> 2<\/span>;<\/span> % 5000 – 2251 + 2 = 2751<\/span><\/p>\n

                                % 4. \u6784\u5efa\u201c\u5e72\u51c0\u201d\u9891\u8c31：\u53ea\u4fdd\u7559\u8fd9\u4e24\u6839\u8c31\u7ebf<\/span>
                                \nX_clean =<\/span> zeros<\/span>(<\/span>1<\/span>,<\/span> N)<\/span>;<\/span>
                                \nX_clean<\/span>(<\/span>k_fund)<\/span> =<\/span> X<\/span>(<\/span>k_fund)<\/span>;<\/span>
                                \nX_clean<\/span>(<\/span>k_neg)<\/span> =<\/span> X<\/span>(<\/span>k_neg)<\/span>;<\/span><\/p>\n

                                % 5. IFFT \u6062\u590d<\/span>
                                \nrecovered_sine =<\/span> real<\/span>(<\/span>ifft<\/span>(<\/span>X_clean)<\/span>)<\/span>;<\/span><\/p>\n

                                % 6. \u7ed8\u56fe\u5bf9\u6bd4<\/span>
                                \nfigure<\/span>(<\/span>5<\/span>)<\/span>;<\/span>
                                \nsubplot<\/span>(<\/span>3<\/span>,<\/span>1<\/span>,<\/span>1<\/span>)<\/span>;<\/span>
                                \nplot<\/span>(<\/span>time<\/span>(<\/span>1<\/span>:<\/span>300<\/span>)<\/span>,<\/span> sin<\/span>(<\/span>2<\/span>*<\/span>pi<\/span>*<\/span>frequency*<\/span>time<\/span>(<\/span>1<\/span>:<\/span>300<\/span>)<\/span>)<\/span>)<\/span>;<\/span>
                                \ntitle<\/span>(<\/span>'\u539f\u59cb\u7eaf\u51c0\u6b63\u5f26\u6ce2 (2.25kHz)'<\/span>)<\/span>;<\/span><\/p>\n

                                subplot<\/span>(<\/span>3<\/span>,<\/span>1<\/span>,<\/span>2<\/span>)<\/span>;<\/span>
                                \nplot<\/span>(<\/span>time<\/span>(<\/span>1<\/span>:<\/span>300<\/span>)<\/span>,<\/span> signal<\/span>(<\/span>1<\/span>:<\/span>300<\/span>)<\/span>)<\/span>;<\/span>
                                \ntitle<\/span>(<\/span>'\u524a\u6ce2\u540e\u7684\u7578\u53d8\u4fe1\u53f7'<\/span>)<\/span>;<\/span><\/p>\n

                                subplot<\/span>(<\/span>3<\/span>,<\/span>1<\/span>,<\/span>3<\/span>)<\/span>;<\/span>
                                \nplot<\/span>(<\/span>time<\/span>(<\/span>1<\/span>:<\/span>300<\/span>)<\/span>,<\/span> recovered_sine<\/span>(<\/span>1<\/span>:<\/span>300<\/span>)<\/span>)<\/span>;<\/span>
                                \ntitle<\/span>(<\/span>'\u6062\u590d\u63d0\u53d6\u7684\u7eaf\u51c0\u57fa\u6ce2 (\u4ec5\u4fdd\u7559 2.25kHz \u9891\u70b9)'<\/span>)<\/span>;<\/span>
                                \nxlabel<\/span>(<\/span>'Time (s)'<\/span>)<\/span>;<\/span><\/p>\n

                                \u6062\u590d\u7ed3\u679c\u5206\u6790\u4e0e\u5e45\u5ea6\u8ba8\u8bba<\/h5>\n
                                  \n
                                • \u6ce2\u5f62\u8d28\u91cf：\u6062\u590d\u51fa\u7684\u6ce2\u5f62\u662f\u5b8c\u7f8e\u5149\u6ed1\u7684\u6b63\u5f26\u6ce2，\u6ca1\u6709\u4efb\u4f55\u201c\u524a\u9876\u201d\u6216\u9636\u68af\u72b6\u5931\u771f\u3002<\/li>\n
                                • \u9891\u7387\u51c6\u786e\u5ea6：\u7cbe\u786e\u4e3a 2.25kHz，\u5206\u6beb\u4e0d\u5dee\u3002<\/li>\n
                                • \u5e45\u5ea6（\u91cd\u70b9）：\u6b64\u65f6 max(recovered_sine) \u7ea6\u7b49\u4e8e 0.7 \u5de6\u53f3（\u7cbe\u786e\u503c\u53d6\u51b3\u4e8e\u524a\u6ce2\u9608\u503c 0.3），\u5e76\u4e0d\u662f\u539f\u59cb\u5e45\u5ea6 1.0\u3002<\/li>\n<\/ul>\n

                                  \u4e3a\u4ec0\u4e48\u5e45\u5ea6\u4e0d\u662f 1.0？
                                  \n\u56e0\u4e3a\u524a\u6ce2\u64cd\u4f5c\u6d88\u8017\u4e86\u57fa\u6ce2\u7684\u80fd\u91cf（\u4e00\u90e8\u5206\u80fd\u91cf\u8f6c\u79fb\u5230\u4e86\u9ad8\u6b21\u8c10\u6ce2\u4e0a）\u3002\u6211\u4eec\u63d0\u53d6\u7684\u53ea\u662f\u201c\u524a\u6ce2\u540e\u5269\u4f59\u7684\u57fa\u6ce2\u5206\u91cf\u201d\u3002\u82e5\u8981\u6062\u590d\u539f\u59cb\u5e45\u5ea6 1.0，\u9700\u8981\u989d\u5916\u77e5\u9053\u524a\u6ce2\u9608\u503c\u5e76\u901a\u8fc7\u67e5\u8868（\u6216\u8fed\u4ee3\u7b97\u6cd5）\u8865\u507f，\u4f46\u4ec5\u9760\u7ebf\u6027\u9891\u57df\u7b5b\u9009\u65e0\u6cd5\u8fd8\u539f\u4e22\u5931\u7684\u80fd\u91cf\u3002<\/p>\n

                                  \n

                                  🔍 \u7b2c\u56db\u90e8\u5206：\u5e38\u89c1\u9519\u8bef\u4ee3\u7801\u6df1\u5ea6\u5256\u6790（\u57fa\u4e8e\u60a8\u63d0\u4f9b\u7684\u6587\u4ef6）<\/h4>\n

                                  \u5728\u60a8\u63d0\u4f9b\u7684 lab1bb.m \u548c lab1bb2.m \u4e2d，\u5b58\u5728\u51e0\u4e2a\u5178\u578b\u9519\u8bef\u3002\u6211\u4eec\u9010\u4e00\u89e3\u5256，\u4ee5\u9632\u8d70\u5165\u8bef\u533a\u3002<\/p>\n

                                  \u9519\u8bef 1：\u9519\u8bef\u5730\u7f6e\u96f6\u4e00\u4e2a\u9891\u7387\u533a\u95f4（lab1bb.m \u4e2d\u7684 Plan2）<\/h5>\n

                                  % \u9519\u8bef\u793a\u4f8b<\/span>
                                  \nclipping_frequency_range =<\/span> [<\/span>1000<\/span>,<\/span> 3000<\/span>]<\/span>;<\/span>
                                  \nclipping_range_indices =<\/span> round<\/span>(<\/span>clipping_frequency_range \/<\/span> frequency_resolution)<\/span> +<\/span> 1<\/span>;<\/span>
                                  \nfrequency_domain_signal<\/span>(<\/span>clipping_range_indices<\/span>(<\/span>1<\/span>)<\/span>:<\/span>clipping_range_indices<\/span>(<\/span>2<\/span>)<\/span>)<\/span> =<\/span> …<\/span>
                                  \n abs<\/span>(<\/span>frequency_domain_signal<\/span>(<\/span>clipping_range_indices<\/span>(<\/span>1<\/span>)<\/span>:<\/span>clipping_range_indices<\/span>(<\/span>2<\/span>)<\/span>)<\/span>)<\/span>;<\/span><\/p>\n

                                  \u4e3a\u4ec0\u4e48\u9519？<\/p>\n

                                    \n
                                  • \u8fd9\u91cc\u8bd5\u56fe\u5c06 1000~3000 Hz \u8303\u56f4\u5185\u7684\u9891\u8c31\u201c\u53d6\u6a21\u201d\u6216\u7f6e\u96f6，\u4f46\u8fd9\u76f8\u5f53\u4e8e\u4e00\u4e2a\u5e26\u963b\u6ee4\u6ce2\u5668\u6216\u5e45\u5ea6\u7be1\u6539，\u4f1a\u7834\u574f\u76f8\u4f4d\u4fe1\u606f，\u4e14\u65e0\u6cd5\u7cbe\u786e\u5206\u79bb\u57fa\u6ce2（\u56e0\u4e3a\u57fa\u6ce2 2250Hz \u5c31\u5728\u8fd9\u4e2a\u8303\u56f4\u5185，\u8fd9\u6837\u505a\u4f1a\u628a\u57fa\u6ce2\u672c\u8eab\u4e5f\u5e72\u6389\u6216\u524a\u5f31）\u3002<\/li>\n
                                  • \u6b63\u786e\u7684\u505a\u6cd5\u662f\u53ea\u4fdd\u7559\u4e00\u4e2a\u9891\u70b9，\u800c\u4e0d\u662f\u4fdd\u7559\u4e00\u4e2a\u9891\u5e26\u3002<\/li>\n<\/ul>\n
                                    \u9519\u8bef 2：\u4f7f\u7528\u524a\u6ce2\u6bd4\u4f8b\u6765\u786e\u5b9a\u7f6e\u96f6\u8303\u56f4（lab1b.m \u4e2d\u7684\u9519\u8bef\u601d\u8def）<\/h5>\n

                                    cut_ratio =<\/span> 0.3<\/span> \/<\/span> 1<\/span>;<\/span> % 0.3<\/span>
                                    \nN_cut =<\/span> round<\/span>(<\/span>N\/<\/span>2<\/span>)<\/span> *<\/span> cut_ratio;<\/span>
                                    \nX_cut<\/span>(<\/span>N_cut+<\/span>1<\/span>:<\/span>end<\/span>)<\/span> =<\/span> 0<\/span>;<\/span><\/p>\n

                                    \u4e3a\u4ec0\u4e48\u9519？<\/p>\n

                                      \n
                                    • \u524a\u6ce2\u6bd4\u4f8b\u51b3\u5b9a\u7684\u662f\u65f6\u57df\u5e45\u5ea6\u622a\u65ad\u7a0b\u5ea6，\u4e0e\u9891\u57df\u8c10\u6ce2\u5206\u5e03\u7684\u4f4d\u7f6e\u65e0\u5173！<\/li>\n
                                    • \u6df7\u53e0\u9891\u7387（\u5982 1.75kHz\u30011.25kHz）\u662f\u7531\u91c7\u6837\u7387\u548c\u4fe1\u53f7\u9891\u7387\u7684\u6570\u5b66\u5173\u7cfb\u51b3\u5b9a\u7684，\u4e0e\u524a\u6ce2\u9608\u503c 0.3 \u6ca1\u6709\u76f4\u63a5\u5173\u7cfb\u3002\u7528 cut_ratio \u53bb\u622a\u65ad\u9891\u8c31\u9ad8\u9891\u7aef，\u76f8\u5f53\u4e8e\u505a\u4f4e\u901a\u6ee4\u6ce2，\u8fd9\u4f1a\u4fdd\u7559\u57fa\u6ce2\u4f46\u4e5f\u4f1a\u4fdd\u7559\u4f4e\u9891\u6df7\u53e0\u6210\u5206（\u5982 0.75kHz），\u6839\u672c\u65e0\u6cd5\u201c\u63d0\u7eaf\u201d\u57fa\u6ce2\u3002<\/li>\n<\/ul>\n
                                      \u9519\u8bef 3：fftshift \u540e\u9519\u8bef\u7f6e\u96f6（lab1bb2.m）<\/h5>\n

                                      y2 =<\/span> fftshift<\/span>(<\/span>abs<\/span>(<\/span>fft<\/span>(<\/span>signal)<\/span>)<\/span>)<\/span>;<\/span>
                                      \ny2<\/span>(<\/span>(<\/span>5001<\/span>–<\/span>1<\/span>)<\/span>*<\/span>0.1<\/span>:<\/span>(<\/span>5001<\/span>–<\/span>1<\/span>)<\/span>*<\/span>0.9<\/span>)<\/span> =<\/span> 0<\/span>;<\/span> % \u8bd5\u56fe\u7f6e\u96f6\u4e2d\u95f4 80% \u7684\u9891\u5e26<\/span><\/p>\n

                                      \u4e3a\u4ec0\u4e48\u9519？<\/p>\n

                                        \n
                                      • \u8fd9\u91cc\u5bf9\u5e45\u5ea6\u8c31（abs）\u8fdb\u884c\u64cd\u4f5c，\u4e22\u5931\u4e86\u76f8\u4f4d\u4fe1\u606f，\u4e14\u7f6e\u96f6\u4e86\u9891\u8c31\u7684\u201c\u4e3b\u4f53\u201d\u90e8\u5206\u3002\u8fd9\u76f8\u5f53\u4e8e\u4e00\u4e2a\u5947\u7279\u7684\u5e26\u963b\u6ee4\u6ce2\u5668，\u5b8c\u5168\u65e0\u6cd5\u8fbe\u5230\u63d0\u53d6\u57fa\u6ce2\u7684\u76ee\u7684\u3002<\/li>\n
                                      • \u66f4\u4e25\u91cd\u7684\u662f，fftshift \u540e\u7684\u7d22\u5f15\u5bf9\u5e94\u7684\u662f -fs\/2 \u5230 fs\/2 \u7684\u987a\u5e8f，\u76f4\u63a5\u7528\u7d22\u5f15\u6bd4\u4f8b\u622a\u65ad\u975e\u5e38\u5371\u9669，\u6781\u6613\u8bef\u4f24\u57fa\u6ce2\u3002<\/li>\n<\/ul>\n
                                        \u6b63\u786e\u65b9\u6cd5\u7684\u672c\u8d28\u603b\u7ed3<\/h5>\n

                                        \u6838\u5fc3\u601d\u60f3：\u5728 DFT \u57df\u4e2d，\u4e0d\u540c\u9891\u7387\u5206\u91cf\u662f\u6b63\u4ea4\u7684\u3002\u5f53\u76ee\u6807\u9891\u7387（\u57fa\u6ce2）\u4e0e\u5e72\u6270\u9891\u7387（\u8c10\u6ce2）\u4f4d\u4e8e\u4e0d\u540c\u7684\u79bb\u6563\u9891\u70b9（bin）\u65f6，\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u201c\u786c\u9009\u9891\u201d（\u53ea\u4fdd\u7559\u76ee\u6807 bin，\u5176\u4f59\u6e05\u96f6）\u5b9e\u73b0\u5b8c\u7f8e\u5206\u79bb\u3002\u8fd9\u79cd\u65b9\u6cd5\u7b49\u6548\u4e8e\u7406\u60f3\u7a84\u5e26\u5e26\u901a\u6ee4\u6ce2\u5668（\u5e26\u5bbd\u7b49\u4e8e\u4e00\u4e2a\u9891\u7387\u5206\u8fa8\u7387）\u3002<\/p>\n

                                        \n

                                        💎 \u603b\u7ed3\u4e0e\u5de5\u7a0b\u5e94\u7528\u542f\u793a<\/h4>\n\n\u5e94\u7528\u573a\u666f\u64cd\u4f5c\u65b9\u6cd5\u6062\u590d\u5e45\u5ea6\u5173\u952e\u6ce8\u610f\u4e8b\u9879<\/tr>\n\n\n\n\n
                                        \u4ece\u8865\u96f6\u7a81\u53d1\u4fe1\u53f7\u63d0\u53d6\u8fde\u7eed\u6b63\u5f26<\/td>\n\u5b9a\u4f4d\u539f\u9891\u7387\u7684\u5171\u8f6d\u9891\u70b9，\u5176\u4f59\u7f6e\u96f6<\/td>\n\u7b49\u4e8e\u539f\u59cb\u5e45\u5ea6（\u672c\u4f8b 1.0）<\/td>\nIFFT \u957f\u5ea6\u4e0e FFT \u957f\u5ea6\u4e00\u81f4\u65f6，\u5e45\u5ea6\u81ea\u7136\u6062\u590d<\/td>\n<\/tr>\n
                                        \u4ece\u524a\u6ce2\u7578\u53d8\u4fe1\u53f7\u63d0\u53d6\u57fa\u6ce2<\/td>\n\u5b9a\u4f4d\u57fa\u6ce2\u9891\u7387\u7684\u5171\u8f6d\u9891\u70b9，\u5176\u4f59\u7f6e\u96f6<\/td>\n\u7b49\u4e8e\u524a\u6ce2\u540e\u7684\u5269\u4f59\u57fa\u6ce2\u5e45\u5ea6（< \u539f\u59cb\u5e45\u5ea6）<\/td>\n\u65e0\u6cd5\u6062\u590d\u4e22\u5931\u7684\u8c10\u6ce2\u80fd\u91cf，\u4ec5\u80fd\u5206\u79bb\u73b0\u6709\u57fa\u6ce2<\/td>\n<\/tr>\n
                                        \u6df7\u53e0\u4fe1\u53f7\u5206\u79bb（\u4e00\u822c\u60c5\u51b5）<\/td>\n\u82e5\u76ee\u6807\u9891\u7387\u4e0e\u6df7\u53e0\u9891\u7387\u4e0d\u5171\u9891\u70b9，\u53ef\u5206\u79bb<\/td>\n\u53d6\u51b3\u4e8e\u8be5\u9891\u7387\u5206\u91cf\u7684\u5b9e\u9645\u80fd\u91cf<\/td>\n\u4e00\u65e6\u4e24\u4e2a\u9891\u7387\u843d\u5728\u540c\u4e00\u4e2a\u9891\u70b9（\u5206\u8fa8\u7387\u4e0d\u591f），\u5219\u6c38\u4e45\u4e0d\u53ef\u5206\u79bb<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                        \u5de5\u7a0b\u542f\u793a：<\/p>\n

                                          \n
                                        • \u8fd9\u79cd\u201c\u9891\u57df\u9009\u9891\u6062\u590d\u201d\u6280\u672f\u5e7f\u6cdb\u5e94\u7528\u4e8e\u7535\u529b\u7cfb\u7edf\u8c10\u6ce2\u5206\u6790（\u63d0\u53d6\u5de5\u9891\u57fa\u6ce2）\u3001\u751f\u7269\u533b\u5b66\u4fe1\u53f7\u5904\u7406（\u63d0\u53d6\u5fc3\u7535\u4fe1\u53f7\u4e2d\u7684\u7279\u5b9a\u8282\u5f8b）\u4ee5\u53ca\u901a\u4fe1\u7cfb\u7edf\u4e2d\u7684\u7a84\u5e26\u5e72\u6270\u6291\u5236\u3002<\/li>\n
                                        • \u4f46\u5b83\u6709\u4e00\u4e2a\u81f4\u547d\u524d\u63d0：\u76ee\u6807\u9891\u7387\u4e0e\u5e72\u6270\u9891\u7387\u5fc5\u987b\u5728\u4e0d\u540c\u7684 FFT \u9891\u70b9\u4e0a\u3002\u82e5\u91c7\u6837\u7387\u8fc7\u4f4e\u5bfc\u81f4\u9891\u7387\u5206\u8fa8\u7387\u592a\u7c97（\u5982 2.25kHz \u4e0e 2.26kHz \u843d\u5728\u4e00\u4e2a bin \u91cc），\u5219\u795e\u4ed9\u96be\u6551\u3002<\/li>\n<\/ul>\n
                                          \n

                                          \u5e0c\u671b\u8fd9\u4efd\u8d85\u8be6\u7ec6\u7684\u7b2c\u4e8c\u7bc7\u6307\u5357，\u914d\u5408\u7b2c\u4e00\u7bc7，\u80fd\u4e3a\u60a8\u6784\u5efa\u8d77\u4ece\u201c\u91c7\u6837\u6df7\u53e0\u201d\u5230\u201c\u9891\u57df\u6062\u590d\u201d\u7684\u5b8c\u6574\u77e5\u8bc6\u95ed\u73af\u3002\u5982\u679c\u60a8\u5bf9\u4ee3\u7801\u4e2d\u7684\u4efb\u4f55\u7ec6\u8282\u8fd8\u6709\u7591\u95ee，\u6b22\u8fce\u7ee7\u7eed\u63a2\u8ba8！
                                          \nWe hope this ultra-detailed second guide, together with the first, builds a complete knowledge loop from \u201csampling aliasing\u201d to \u201cfrequency-domain recovery.\u201d Feel free to discuss any remaining questions about the code! 😊<\/p>\n","protected":false},"excerpt":{"rendered":"

                                          \u8d85\u8be6\u7ec6\u6307\u5357：MATLAB \u9891\u57df\u64cd\u63a7\u4e0e\u4fe1\u53f7\u6062\u590d \u2014\u2014 \u4ece\u96f6\u586b\u5145\u4fe1\u53f7\u5230\u524a\u6ce2\u57fa\u6ce2\u63d0\u53d6Ultra-Detailed Guide: Frequency-Domain Manipulation and Signal Recovery in MATLAB \u2014 From Zero-Padded Signals to Fundamental Extraction from Clipped Waveforms🌐 English Version (Ultra-Deta<\/p>\n","protected":false},"author":2,"featured_media":94064,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[1284,2710,190],"topic":[],"class_list":["post-94066","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-server","tag-matlab","tag-2710","tag-190"],"yoast_head":"\n\u3010\u9ad8\u7ea7\u6570\u5b57\u4fe1\u53f7\u5904\u7406\u3011\u8d85\u8be6\u7ec6\u6307\u5357lab1b\uff1aMATLAB \u9891\u57df\u64cd\u63a7\u4e0e\u4fe1\u53f7\u6062\u590d \u2014\u2014 \u4ece\u96f6\u586b\u5145\u4fe1\u53f7\u5230\u524a\u6ce2\u57fa\u6ce2\u63d0\u53d6\u3010\u542bmatlab\u4ee3\u7801\u3011 - 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