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Transformer has obtained promising results on cognitive speech signal processing field, which is of interest in various applications ranging from emotion to neurocognitive disorder analysis. However, most works treat speech signal as a whole, leading to the neglect of the pronunciation structure that is unique to speech and reflects the cognitive process. Meanwhile, Transformer has heavy computational burden due to its full attention operation. In this paper, a hierarchical efficient framework, called SpeechFormer, which considers the structural characteristics of speech, is proposed and can be served as a general-purpose backbone for cognitive speech signal processing. The proposed SpeechFormer consists of frame, phoneme, word and utterance stages in succession, each performing a neighboring attention according to the structural pattern of speech with high computational efficiency. SpeechFormer is evaluated on speech emotion recognition (IEMOCAP & MELD) and neurocognitive disorder detection (Pitt & DAIC-WOZ) tasks, and the results show that SpeechFormer outperforms the standard Transformer-based framework while greatly reducing the computational cost. Furthermore, our SpeechFormer achieves comparable results to the state-of-the-art approaches.